How Often to Sample a Continuous-Time
Process in the Presence of Market
Microstructure Noise
Yacine Aı̈t-Sahalia
Princeton University and NBER
Per A. Mykland
The University of Chicago
Lan Zhang
Carnegie Mellon University
In theory, the sum of squares of log returns sampled at high frequency estimates their
variance. When market microstructure noise is present but unaccounted for, however,
we show that the optimal sampling frequency is finite and derives its closed-form
expression. But even with optimal sampling, using say 5-min returns when transactions are recorded every second, a vast amount of data is discarded, in contradiction
to basic statistical principles. We demonstrate that modeling the noise and using all
the data is a better solution, even if one misspecifies the noise distribution. So the
answer is: sample as often as possible.
Over the past few years, price data sampled at very high frequency have
become increasingly available in the form of the Olsen dataset of currency
exchange rates or the TAQ database of NYSE stocks. If such data were
not affected by market microstructure noise, the realized volatility of the
process (i.e., the average sum of squares of log-returns sampled at high
frequency) would estimate the returns’ variance, as is well known. In fact,
sampling as often as possible would theoretically produce in the limit a
perfect estimate of that variance.
We start by asking whether it remains optimal to sample the price
process at very high frequency in the presence of market microstructure
noise, consistently with the basic statistical principle that, ceteris paribus,
more data are preferred to less. We first show that, if noise is present but
unaccounted for, then the optimal sampling frequency is finite, and we
We are grateful for comments and suggestions from the editor, Maureen O’Hara, and two anonymous
referees, as well as seminar participants at Berkeley, Harvard, NYU, MIT, Stanford, the Econometric
Society and the Joint Statistical Meetings. Financial support from the NSF under grants SBR-0111140
(Aı̈t-Sahalia), DMS-0204639 (Mykland and Zhang), and the NIH under grant RO1 AG023141-01
(Zhang) is also gratefully acknowledged. Address correspondence to: Yacine Aı̈t-Sahalia, Bendheim Center
for Finance, Princeton University, Princeton, NJ 08540, (609) 258-4015 or email: [email protected].
The Review of Financial Studies Vol. 18, No. 2 ª 2005 The Society for Financial Studies; all rights reserved.
doi:10.1093/rfs/hhi016
Advance Access publication February 10, 2005
The Review of Financial Studies / v 18 n 2 2005
derive a closed-form formula for it. The intuition for this result is as
follows. The volatility of the underlying efficient price process and the
market microstructure noise tend to behave differently at different frequencies. Thinking in terms of signal-to-noise ratio, a log-return observed
from transaction prices over a tiny time interval is mostly composed of
market microstructure noise and brings little information regarding the
volatility of the price process since the latter is (at least in the Brownian
case) proportional to the time interval separating successive observations.
As the time interval separating the two prices in the log-return increases,
the amount of market microstructure noise remains constant, since each
price is measured with error, while the informational content of volatility
increases. Hence very high frequency data are mostly composed of market
microstructure noise, while the volatility of the price process is more
apparent in longer horizon returns. Running counter to this effect is the
basic statistical principle mentioned above: in an idealized setting where
the data are observed without error, sampling more frequently can be
useful. What is the right balance to strike? What we show is that these two
effects compensate each other and result in a finite optimal sampling
frequency (in the root mean squared error sense) so that some time
aggregation of the returns data is advisable.
By providing a quantitative answer to the question of how often one
should sample, we hope to reduce the arbitrariness of the choices that have
been made in the empirical literature using high frequency data: for
example, using essentially the same Olsen exchange rate series, these
somewhat ad hoc choices range from 5-min intervals [e.g., Andersen
et al. (2001), Barndorff-Nielsen and Shephard (2002), Gençay et al.
(2002) to as long as 30 min [e.g., Andersen et al. (2003)]. When calibrating
our analysis to the amount of microstructure noise that has been reported
in the literature, we demonstrate how the optimal sampling interval
should be determined: for instance, depending upon the amount of microstructure noise relative to the variance of the underlying returns, the
optimal sampling frequency varies from 4 min to 3 h, if 1 day’s worth of
data are used at a time. If a longer time period is used in the analysis, then
the optimal sampling frequency can be considerably longer than these
values.
But even if one determines the sampling frequency optimally, the fact
remains that the empirical researcher is not making full use of the data at
his disposal. For instance, suppose that we have available transaction
records on a liquid stock, traded once every second. Over a typical 6.5 h
day, we therefore start with 23,400 observations. If one decides to sample
once every 5 minutes, then — whether or not this is the optimal sampling
frequency — it amounts to retaining only 78 observations. Stated differently, one is throwing away 299 out of every 300 transactions. From a
statistical perspective, this is unlikely to be the optimal solution, even
352
Sampling and Market Microstructure Noise
though it is undoubtedly better than computing a volatility estimate using
noisy squared log-returns sampled every second. Somehow, an optimal
solution should make use of all the data, and this is where our analysis is
headed next.
So, if one decides to account for the presence of the noise, how should
one proceed? We show that modeling the noise term explicitly restores
the first order statistical effect that sampling as often as possible is optimal. This will involve an estimator different from the simple sum of
squared log-returns. Since we work within a fully parametric framework,
likelihood is the key word. Hence we construct the likelihood function for
the observed log-returns, which include microstructure noise. To do so, we
must postulate a model for the noise term. We assume that the noise is
Gaussian. In light of what we know from the sophisticated theoretical
microstructure literature, this is likely to be overly simplistic and one may
well be concerned about the effect(s) of this assumption. Could it be more
harmful than useful? Surprisingly, we demonstrate that our likelihood
correction, based on the Gaussianity of the noise, works even if one
misspecifies the assumed distribution of the noise term. Specifically, if
the econometrician assumes that the noise terms are normally distributed
when in fact they are not, not only is it still optimal to sample as often as
possible (unlike the result when no allowance is made for the presence of
noise), but the estimator has the same variance as if the noise distribution
had been correctly specified. This robustness result is, we think, a major
argument in favor of incorporating the presence of the noise when estimating continuous time models with high frequency financial data, even if
one is unsure about the true distribution of the noise term.
In other words, the answer to the question we pose in our title is ‘‘as
often as possible,’’ provided one accounts for the presence of the noise
when designing the estimator (and we suggest maximum likelihood as a
means of doing so). If one is unwilling to account for the noise, then one
has to rely on the finite optimal sampling frequency we start our analysis
with. However, we stress that while it is optimal if one insists upon using
sums of squares of log-returns, this is not the best possible approach to
estimate volatility given the complete high frequency dataset at hand.
In a companion paper [Zhang, Mykland, and Aı̈t-Sahalia (2003)], we
study the corresponding nonparametric problem, where the volatility of
the underlying price is a stochastic process, and nothing else is known
about it, in particular no parametric structure. In that case, the object of
interest is the integrated volatility of the process over a fixed time interval,
such as a day, and we show how to estimate it using again all the data
available (instead of sparse sampling at an arbitrarily lower frequency of,
say, 5 min). Since the model is nonparametric, we no longer use a likelihood approach but instead propose a solution based on subsampling and
averaging, which involves estimators constructed on two different time
353
The Review of Financial Studies / v 18 n 2 2005
scales, and demonstrate that this again dominates sampling at a lower
frequency, whether arbitrarily or optimally determined.
This article is organized as follows. We start by describing in Section 1
our reduced form setup and the underlying structural models that support
it. We then review in Section 2 the base case where no noise is present,
before analyzing in Section 3 the situation where the noise is ignored. In
Section 4, we examine the concrete implications of this result for empirical
work with high frequency data. Next, we show in Section 5 that accounting for the presence of the noise through the likelihood restores the
optimality of high frequency sampling. Our robustness results are presented in Section 6 and interpreted in Section 7. In Section 8, we study the
same questions when the observations are sampled at random time intervals, which are an essential feature of transaction-level data. We then turn
to various extensions and relaxation of our assumptions in Section 9; we
added a drift term, then serially correlated and cross-correlated noise
respectively. In Section 10 concludes. All proofs are in the appendix.
1. Setup
Our basic setup is as follows. We assume that the underlying process of
interest, typically the log-price of a security, is a time-homogeneous
diffusion on the real line
dXt ¼ mðXt ; uÞdt þ sdWt ,
ð1Þ
where X0 ¼ 0, Wt is a Brownian motion, m(., .) is the drift function, s2 the
diffusion coefficient, and u the drift parameters, where u 2 Q and s > 0.
The parameter space is an open and bounded set. As usual, the restriction
that s is constant is without loss of generality since in the univariate case a
one-to-one transformation can always reduce a known specification s(Xt)
to that case. Also, as discussed in Aı̈t-Sahalia and Mykland (2003), the
properties of parametric estimators in this model are quite different
depending upon whether we estimate u alone, s2 alone, or both parameters together. When the data are noisy, the main effects that we describe
are already present in the simpler of these three cases, where s2 alone is
estimated, and so we focus on that case. Moreover, in the high frequency
context we have in mind, the diffusive component of (1) is of order (dt)1/2
while the drift component is of order dt only, so the drift component is
mathematically negligible at high frequencies. This is validated empirically: including a drift which actually deteriorates the performance of variance estimates from high frequency data since the drift is estimated with a
large standard error. Not centering the log returns for the purpose of
variance estimation produces more accurate results [see Merton (1980)].
So we simplify the analysis one step further by setting m ¼ 0, which we do
354
Sampling and Market Microstructure Noise
until Section 9.1, where we then show that adding a drift term does
not alter our results. In Section 9.4, we discuss the situation where the
instantaneous volatility s is stochastic.
But for now,
Xt ¼ sWt :
ð2Þ
Until Section 8, we treat the case where our observations occur at equidistant time intervals D, in which case the parameter s2 is estimated at
time T on the basis of N þ 1 discrete observations recorded at times t 0 ¼ 0,
t 1 ¼ D, . . . ,t N ¼ N D ¼ T. In Section 8, we let the sampling intervals themselves be random variables, since this feature is an essential characteristic
of high frequency transaction data.
The notion that the observed transaction price in high frequency financial data is the unobservable efficient price plus some noise component
due to the imperfections of the trading process is a well established
concept in the market microstructure literature [see, for instance Black
(1986)]. So, we depart from the inference setup previously studied
[Aı̈t-Sahalia and Mykland (2003)] and we now assume that, instead of
observing the process X at dates t i, we observe X with error:
~ ti ¼ Xti þ Uti ,
X
ð3Þ
where the Utis are i.i.d. noise with mean zero and variance a2 and are
independent of the W process. In this context, we view X as the efficient
~ is the transaction log-price. In an efficient
log-price, while the observed X
market, Xt is the log of the expectation of the final value of the security
conditional on all publicly available information at time t. It corresponds
to the log-price that would be, in effect, in a perfect market with no trading
imperfections, frictions, or informational effects. The Brownian motion
W is the process representing the arrival of new information, which in this
idealized setting is immediately impounded in X.
By contrast, Ut summarizes the noise generated by the mechanics of the
trading process. We view the source of noise as a diverse array of market
microstructure effects, either information or non-information related,
such as the presence of a bid-ask spread and the corresponding bounces,
the differences in trade sizes and the corresponding differences in representativeness of the prices, the different informational content of price
changes owing to informational asymmetries of traders, the gradual
response of prices to a block trade, the strategic component of the order
flow, inventory control effects, the discreteness of price changes in markets that are not decimalized, etc., all summarized into the term U. That
these phenomena are real and important and this is an accepted fact in the
market microstructure literature, both theoretical and empirical. One can
in fact argue that these phenomena justify this literature.
355
The Review of Financial Studies / v 18 n 2 2005
We view Equation (3) as the simplest possible reduced form of structural market microstructure models. The efficient price process X is typically modeled as a random walk, that is, the discrete time equivalent of
Equation (2). Our specification coincides with that of Hasbrouck (1993),
who discusses the theoretical market microstructure underpinnings of
such a model and argues that the parameter a is a summary measure of
market quality. Structural market microstructure models do generate
Equation (3). For instance, Roll (1984) proposes a model where U is due
entirely to the bid-ask spread. Harris (1990b) notes that in practice there
are sources of noise other than just the bid-ask spread, and studies their
effect on the Roll model and its estimators.
Indeed, a disturbance U can also be generated by adverse selection
effects as in Glosten (1987) and Glosten and Harris (1988), where the
spread has two components: one that is owing to monopoly power, clearing costs, inventory carrying costs, etc., as previously, and a second one
that arises because of adverse selection whereby the specialist is concerned
that the investor on the other side of the transaction has superior information. When asymmetric information is involved, the disturbance U
would typically no longer be uncorrelated with the W process and would
exhibit autocorrelation at the first order, which would complicate our
analysis without fundamentally altering it: see Sections 9.2 and 9.3
where we relax the assumptions that the Us are serially uncorrelated and
independent of the W process.
The situation where the measurement error is primarily due to the fact
~ ti ¼ mi k where k is
that transaction prices are multiples of a tick size (i.e., X
the tick size and mi is the integer closest to Xti /k) can be modeled as a
rounding off problem [see Gottlieb and Kalay (1985), Jacod (1996),
Delattre and Jacod (1997)]. The specification of the model in Harris
(1990a) combines both the rounding and bid-ask effects as the dual
sources of the noise term U. Finally, structural models, such as that of
Madhavan, Richardson, and Roomans (1997), also give rise to reduced
~ takes the form of an unobforms where the observed transaction price X
served fundamental value plus error.
With Equation (3) as our basic data generating process, we now turn
to the questions we address in this article: how often should one sample
a continuous-time process when the data are subject to market microstructure noise, what are the implications of the noise for the estimation
of the parameters of the X process, and how should one correct for the
presence of the noise, allowing for the possibility that the econometrician
misspecifies the assumed distribution of the noise term, and finally
allowing for the sampling to occur at random points in time? We proceed from the simplest to the most complex situation by adding one
extra layer of complexity at a time: Figure 1 shows the three sampling
schemes we consider, starting with fixed sampling without market
356
Sampling and Market Microstructure Noise
Figure 1
Various discrete sampling modes — no noise (Section 2), with noise (Sections 3–7) and randomly spaced with
noise (Section 8)
microstructure noise, then moving to fixed sampling with noise and
concluding with an analysis of the situation where transaction prices
are not only subject to microstructure noise but are also recorded at
random time intervals.
357
The Review of Financial Studies / v 18 n 2 2005
2. The Baseline Case: No Microstructure Noise
We start by briefly reviewing what would happen in the absence of
market microstructure noise, that is when a ¼ 0. With X denoting the
log-price, the first differences of the observations are the log-returns
~ ti X
~ ti1 , i ¼ 1, . . . , N. The observations Yi ¼ s(Wt Wt ) are
Yi ¼ X
iþ1
i
then i.i.d. N(0, s2D) so the likelihood function is
1
ð4Þ
l s2 ¼ N ln 2ps2 D =2 2s2 D Y 0 Y ,
where Y ¼ (Y1, . . . ,YN)0 . The maximum-likelihood estimator of s2 coincides with the discrete approximation to the quadratic variation of the
process
s
^2 ¼
N
1X
Y 2,
T i¼1 i
ð5Þ
which has the following exact small sample moments:
N
2 1 X
2 N s2 D
¼ s2 ,
E Yi ¼
E s
^ ¼
T
T i¼1
"
#
!
N
N
X
X
2
2s4 D
1
1
N
var s
^ ¼ 2 var
Yi2 ¼ 2
var Yi2
¼ 2 2s4 D2 ¼
T
T
T
T
i¼1
i¼1
and the following asymptotic distribution
2
^ s2 ! N ð0, vÞ,
T 1=2 s
T!1
ð6Þ
where
h i1
2
¼ 2s4 D:
v ¼ avar s
^ ¼ DE €l s2
ð7Þ
Thus selecting D as small as possible is optimal for the purpose of
estimating s2.
3. When the Observations are Noisy but the Noise is Ignored
Suppose now that market microstructure noise is present but the presence
of the Us is ignored when estimating s2. In other words, we use the
log-likelihood function (4) even though the true structure of the observed
log-returns Yis is given by an MA(1) process since
~ ti X
~ ti1
Yi ¼ X
¼ Xti Xti1 þ Uti Uti1
¼ sðWti Wti1 Þ þ Uti Uti1
«i þ h«i1 ,
358
ð8Þ
Sampling and Market Microstructure Noise
where the «is are uncorrelated with mean zero and variance g2 (if the Us
are normally distributed, then the «is are i.i.d.). The relationship to the
original parametrization (s2, a2) is given by
g 2 1 þ h2 ¼ var½Yi ¼ s2 D þ 2a2 ,
ð9Þ
g 2 h ¼ covðYi , Yi1 Þ ¼ a2 :
ð10Þ
Equivalently, the inverse change of variable is given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2a2 þ s2 D þ s2 Dð4a2 þ s2 DÞ ,
2
ð11Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2
2
h ¼ 2 2a s D þ s2 Dð4a2 þ s2 DÞ :
2a
ð12Þ
g2 ¼
Two important properties of the log-returns Yi s emerge from Equations
(9) and (10). First, it is clear from Equation (9) that microstructure noise
leads to spurious variance in observed log-returns, s2D þ 2a2 versus s2D.
This is consistent with the predictions of theoretical microstructure
models. For instance, Easley and O’Hara (1992) develop a model linking
the arrival of information, the timing of trades, and the resulting price
process. In their model, the transaction price will be a biased representation of the efficient price process, with a variance that is both overstated
and heteroskedastic due to the fact that transactions (hence the recording
~ ) occur at intervals that are timeof an observation on the process X
varying. While our specification is too simple to capture the rich joint
dynamics of price and sampling times predicted by their model, heteroskedasticity of the observed variance will also appear in our case
once we allow for time variation of the sampling intervals (see Section 8).
In our model, the proportion of the total return variance that is market
microstructure-induced is
p¼
2a2
þ 2a2
s2 D
ð13Þ
at observation interval D. As D gets smaller, p gets closer to 1, so that a
larger proportion of the variance in the observed log-return is driven by
market microstructure frictions, and correspondingly a lesser fraction
reflects the volatility of the underlying price process X.
Second, Equation (10) implies that 1 < h < 0, so that logreturns are (negatively) autocorrelated with first order autocorrelation
a2/(s2D þ 2a2) ¼ p/2. It has been noted that market microstructure
noise has the potential to explain the empirical autocorrelation of returns.
For instance, in the simple Roll model, Ut ¼ (s/2)Qt where s is the bid/ask
359
The Review of Financial Studies / v 18 n 2 2005
spread and Qt, the order flow indicator, is a binomial variable that takes
the values þ1 and 1 with equal probability. Therefore, var[Ut] ¼ a2 ¼ s2/4.
Since cov(Yi,Yi 1) ¼ a2, the bid/ask spread can be recovered in this
pffiffiffiffiffiffiffi
model as s ¼ 2 r where r ¼ g2h is the first-order autocorrelation of
returns. French and Roll (1986) proposed to adjust variance estimates to
control for such autocorrelation and Harris (1990b) studied the resulting
estimators. In Sias and Starks (1997), U arises because of the strategic
trading of institutional investors which is then put forward as an explanation for the observed serial correlation of returns. Lo and MacKinlay
(1990) show that infrequent trading has implications for the variance and
autocorrelations of returns. Other empirical patterns in high frequency
financial data have been documented: leptokurtosis, deterministic
patterns, and volatility clustering.
Our first result shows that the optimal sampling frequency is finite when
noise is present but unaccounted for. The estimator s
^ 2 obtained from
maximizing the misspecified log-likelihood function (4) is quadratic in the
Yi s [see Equation (5)]. In order to obtain its exact (i.e., small sample)
variance, we need to calculate the fourth order cumulants of the Yi s since
2
cov Yi2 , Yj2 ¼ 2 cov Yi , Yj þ cum Yi , Yi , Yj , Yj
ð14Þ
(see, e.g., Section 2.3 of McCullagh (1987) for definitions and properties of
the cumulants). We have the following lemma.
Lemma 1. The fourth cumulants of the log-returns are given by
cum Yi ,Yj ,Yk , Yl
8
if i ¼ j ¼ k ¼ l,
>
< 2cum4 ½U ,
sði;j;k;l Þ
¼ ð1Þ
cum4 ½U , if maxði, j,k, l Þ ¼ minði, j, k, l Þ þ 1,
>
:
0,
otherwise,
ð15Þ
where s(i, j, k, l) denotes the number of indices among (i, j, k, l) that are
equal to min(i, j, k, l) and U denotes a generic random variable with the
common distribution of the Utis. Its fourth cumulant is denoted cum4 [U].
Now U has mean zero, so in terms of its moments
2
cum4 ½U ¼ E U 4 3 E U 2 :
ð16Þ
In the special case where U is normally distributed, cum4 [U] ¼ 0 and as a
result of Equation (14) the fourth cumulants of the log-returns are all 0
(since W is normal, the log-returns are also normal in that case). If the
distribution of U is binomial as in the simple bid/ask model described
above, then cum4 [U ] ¼ s4/8; since in general s will be a tiny percentage
of the asset price, say s ¼ 0.05%, the resulting cum4 [U ] will be very small.
360
Sampling and Market Microstructure Noise
We can now characterize the root mean squared error
2 2 2
2 1=2
^ s2 þ var s
^
RMSE s
^ ¼ E s
of the estimator by the following theorem.
Theorem 1. In small samples (finite T), the bias and variance of the
estimator s
^ 2 are given by
2
2a2
,
E s
^ s2 ¼
D
var s
^
2
ð17Þ
2 s4 D2 þ 4s2 Da2 þ 6a4 þ 2 cum4 ½U 2 2a4 þ cum4 ½U ¼
:
TD
T2
ð18Þ
Its RMSE has a unique minimum in D which is reached at the optimal
sampling interval
00
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11=3
3
4 1=3
2 3a4 þ cum4 ½U 2a
T
B@
A
D ¼
@ 1 1
s4
27s4 a8 T 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11=3 1
3
2 3a4 þ cum4 ½U @
A C
þ 1þ 1
A:
27s4 a8 T 2
0
As T grows, we have
22=3 a4=3 1=3
1
D ¼
T þ O 1=3 :
T
s4=3
ð19Þ
ð20Þ
The trade-off between bias and variance made explicit in Equations (17)
and (18) is not unlike the situation in nonparametric estimation with D1
playing the role of the bandwidth h. A lower h reduces the bias but
increases the variance, and the optimal choice of h balances the two
effects.
Note that these are exact small sample expressions, valid for all T.
Asymptotically in T, var½^
s2 ! 0, and hence the RMSE of the estimator
is dominated by the bias term which is independent of T. And given the
form of the bias (17), one would in fact want to select the largest D possible
to minimize the bias (as opposed to the smallest one as in the no-noise case
of Section 2). The rate at which D should increase with T is given by
Equation (20). Also, in the limit where the noise disappears (a ! 0 and
cum4 [U ] ! 0), the optimal sampling interval D tends to 0.
361
The Review of Financial Studies / v 18 n 2 2005
How does a small departure from a normal distribution of the microstructure noise affect the optimal sampling frequency? The answer is that a
small positive (resp. negative) departure of cum4[U] starting from the
normal value of 0 leads to an increase (resp. decrease) in D, since
0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2=3
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2=3 1
4
2a4
@ 1 þ 1 2a
A
1 1 2 4
2
4
T s
T s
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cum4 ½U D ¼ Dnormal þ
2a4 8=3
1=3
4=3
1=3
32 a T
1 2 4s
T s
þ O cum4 ½U 2 ,
ð21Þ
where Dnormal is the value of D corresponding to Cum4 [U] ¼ 0. And, of
course, the full formula (19) can be used to get the exact answer for any
departure from normality instead of the comparative static one.
Another interesting asymptotic situation occurs if one attempts to use
higher and higher frequency data (D ! 0, say sampled every minute) over
a fixed time period (T fixed, say a day). Since the expressions in Theorem 1
are exact small sample ones, they can in particular be specialized to
analyze this situation. With n ¼ T/D, it follows from Equations (17) and
(18) that
2 2na2
2nE U 2
E s
^ ¼
þ oð nÞ ¼
þ oðnÞ,
T
T
2 2n 6a4 þ 2 cum4 ½U 4nE U 4
var s
^ ¼
þ
o
ð
n
Þ
¼
þ oð nÞ
T2
T2
ð22Þ
ð23Þ
so ðT=2nÞ^
s2 becomes an estimator of E [U2] ¼ a2 whose asymptotic variance is E [U 4]. Note in particular that s
^ 2 estimates the variance of the
noise, which is essentially unrelated to the object of interest s2. This type
of asymptotics is relevant in the stochastic volatility case we analyze in our
companion paper [Zhang, Mykland, and Aı̈t-Sahalia (2003)].
Our results also have implications for the two parallel tracks that have
developed in the recent financial econometrics literature dealing with
discretely observed continuous-time processes. One strand of the literature has argued that estimation methods should be robust to the potential
issues arising in the presence of high frequency data and, consequently, be
asymptotically valid without requiring that the sampling interval D
separating successive observations tend to zero [see, e.g., Hansen and
Scheinkman (1995), Aı̈t-Sahalia (1996), Aı̈t-Sahalia (2002)]. Another
strand of the literature has dispensed with that constraint, and the asymptotic validity of these methods requires that D tend to zero instead of or in
addition to, an increasing length of time T over which these observations
362
Sampling and Market Microstructure Noise
are recorded [see, e.g., Andersen et al. (2003), Bandi and Phillips (2003),
Barndorff-Nielsen and Shephard (2002)].
The first strand of literature has been informally warning about the
potential dangers of using high frequency financial data without accounting for their inherent noise [see, e.g., Aı̈t-Sahalia (1996, p. 529)], and we
propose a formal modelization of that phenomenon. The implications of
our analysis are most important for the second strand of the literature,
which is predicated on the use of high frequency data but does not account
for the presence of market microstructure noise. Our results show that the
properties of estimators based on the local sample path properties of the
process (such as the quadratic variation to estimate s2) change dramatically in the presence of noise. Complementary to this are the results of
Gloter and Jacod (2000) which show that the presence of even increasingly
negligible noise is sufficient to adversely affect the identification of s2.
4. Concrete Implications for Empirical Work with High Frequency Data
The clear message of Theorem 1 for empirical researchers working with
high frequency financial data is that it may be optimal to sample less
frequently. As discussed in the Introduction, authors have reduced their
sampling frequency below that of the actual record of observations in a
somewhat ad hoc fashion, with typical choices 5 min and up. Our analysis
provides not only a theoretical rationale for sampling less frequently, but
also gives a precise answer to the question of ‘‘how often one should
sample?’’ For that purpose, we need to calibrate the parameters appearing
in Theorem 1, namely s, a, cum4[U], D and T. We assume in this calibration exercise that the noise is Gaussian, in which case cum4[U ] ¼ 0.
4.1 Stocks
We use existing studies in empirical market microstructure to calibrate the
parameters. One such study is Madhavan, Richardson, and Roomans
(1997), who estimated on the basis of a sample of 274 NYSE stocks that
approximately 60% of the total variance of price changes is attributable to
market microstructure effects (they report a range of values for p from
54% in the first half hour of trading to 65% in the last half hour, see their
Table 4; they also decompose this total variance into components due to
discreteness, asymmetric information, transaction costs and the interaction between these effects). Given that their sample contains an average of
15 transactions per hour (their Table 1), we have in our framework
p ¼ 60%,
D ¼ 1=ð15 7 252Þ:
ð24Þ
These values imply from Equation (13) that a ¼ 0.16% if we assume a
realistic value of s ¼ 30% per year. (We do not use their reported
363
The Review of Financial Studies / v 18 n 2 2005
Table 1
Optimal sampling frequency
T
Value of a
1 day
0.01%
0.05%
0.1%
0.15%
0.2%
0.3%
0.4%
0.5%
0.6%
0.7%
0.8%
0.9%
1.0%
1 min
5 min
12 min
22 min
32 min
57 min
1.4 h
2h
2.6 h
3.3 h
4.1 h
4.9 h
5.9 h
1 year
5 years
4 min
31 min
1.3 h
2.2 h
3.3 h
5.6 h
1.3 day
1.7 day
2.2 days
2.7 days
3.2 days
3.8 days
4.3 days
6 min
53 min
2.2 h
3.8 h
5.6 h
1.5 day
2.2 days
2.9 days
3.7 days
4.6 days
1.1 week
1.3 week
1.5 week
(a) s ¼ 30% Stocks
(b) s ¼ 10% Currencies
0.005%
0.01%
0.02%
0.05%
0.10%
4 min
9 min
23 min
1.3 h
3.5 h
23 min
58 min
2.4 h
8.2 h
20.7 h
39 min
1.6 h
4.1 h
14.0 h
1.5 day
This table reports the optimal sampling frequency D given in Equation 19 for different values of the
standard deviation of the noise term a and the length of the sample T. Throughout the table, the noise
is assumed to be normally distributed (hence cum4[U] ¼ 0 in formula 19). In panel (a), the standard
deviation of the efficient price process is s ¼ 30% per year, and at s ¼ 10% per year in panel (b). In both
panels, 1 year ¼ 252 days, but in panel (a), 1 day ¼ 6.5 hours (both the NYSE and NASDAQ are open for
6.5 h from 9:30 to 16:00 EST), while in panel (b), 1 day ¼ 24 hours as is the case for major currencies. A
value of a ¼ 0.05% means that each transaction is subject to Gaussian noise with mean 0 and standard
deviation equal to 0.05% of the efficient price. If the sole source of the noise were a bid/ask spread of size s,
then a should be set to s/2. For example, a bid/ask spread of 10 cents on a $10 stock would correspond to
a ¼ 0.05%. For the dollar/euro exchange rate, a bid/ask spread of s ¼ 0.04% translates into a ¼ 0.02%. For
the bid/ask model, which is based on binomial instead of Gaussian noise, cum4[U] ¼ s4/8, but this
quantity is negligible given the tiny size of s.
volatility number since they apparently averaged the variance of price
changes over the 274 stocks instead of the variance of the returns. Since
different stocks have different price levels, the price variances across
stocks are not directly comparable. This does not affect the estimated
fraction p, however, since the price level scaling factor cancels out
between the numerator and the denominator.)
The magnitude of the effect is bound to vary by type of security, market
and time period. Hasbrouck (1993) estimates the value of a to be 0.33%.
Some authors have reported even larger effects. Using a sample of NASDAQ stocks, Kaul and Nimalendran (1990) estimate that about 50% of
the daily variance of returns is due to the bid-ask effect. With s ¼ 40%
(NASDAQ stocks have higher volatility), the values
p ¼ 50%,
364
D ¼ 1=252
Sampling and Market Microstructure Noise
yield the value a ¼ 1.8%. Also on NASDAQ, Conrad, Kaul and
Nimalendran (1991) estimate that 11% of the variance of weekly returns
(see their Table 4, middle portfolio) is due to bid-ask effects. The values
p ¼ 11%,
D ¼ 1=52
imply that a ¼ 1.4%.
In Table 1, we compute the value of the optimal sampling interval D
implied by different combinations of sample length (T ) and noise magnitude (a). The volatility of the efficient price process is held fixed at
s ¼ 30% in Panel (A), which is a realistic value for stocks. The numbers
in the table show that the optimal sampling frequency can be substantially
affected by even relatively small quantities of microstructure noise.
For instance, using the value a ¼ 0.15% calibrated from Madhavan,
Richardson, and Roomans (1997), we find an optimal sampling interval
of 22 minutes if the sampling length is 1 day; longer sample lengths lead to
higher optimal sampling intervals. With the higher value of a ¼ 0.3%,
approximating the estimate from Hasbrouck (1993), the optimal sampling
interval is 57 min. A lower value of the magnitude of the noise translates
into a higher frequency: for instance, D ¼ 5 min if a ¼ 0.05% and
T ¼ 1 day. Figure 2 displays the RMSE of the estimator as a function of
D and T, using parameter values s ¼ 30% and a ¼ 0.15%. The figure
illustrates the fact that deviations from the optimal choice of D lead to a
substantial increase in the RMSE: for example, with T ¼ 1 month, the
RMSE more than doubles if, instead of the optimal D ¼ 1 h, one uses
D ¼ 15 min.
Figure 2
^ 2 when the presence of the noise is ignored
RMSE of the estimator s
365
The Review of Financial Studies / v 18 n 2 2005
Table 2
Monte Carlo simulations: bias and variance when market microstructure noise is ignored
Sampling interval
5 min
15 min
30 min
1h
2h
1 day
1 week
Theoretical mean
Sample mean
Theoretical stand. dev.
Sample stand. dev.
0.185256
0.121752
0.10588
0.097938
0.09397
0.09113
0.0902
0.185254
0.121749
0.10589
0.097943
0.09401
0.09115
0.0907
0.00192
0.00208
0.00253
0.00330
0.00448
0.00812
0.0177
0.00191
0.00209
0.00254
0.00331
0.00440
0.00811
0.0176
This table reports the results of M ¼ 10,000 Monte Carlo simulations of the estimator s
^ 2 , with market
microstructure noise present but ignored. The column ‘‘theoretical mean’’ reports the expected value of the
estimator, as given in Equation (17) and similarly for the column ‘‘theoretical standard deviation’’ (the
variance is given in Equation (18)). The ‘‘sample’’ columns report the corresponding moments computed
over the M simulated paths. The parameter values used to generate the simulated data are s2 ¼ 0.32 ¼ 0.09
and a2 ¼ (0.15%)2 and the length of each sample is T ¼ 1 year.
4.2 Currencies
Looking now at foreign exchange markets, empirical market microstructure studies have quantified the magnitude of the bid-ask spread. For
example, Bessembinder (1994) computes the average bid/ask spread s in
the wholesale market for different currencies and reports values of
s ¼ 0.05% for the German mark, and 0.06% for the Japanese yen (see
Panel B of his Table 2). We calculated the corresponding numbers for the
1996–2002 period to be 0.04% for the mark (followed by the euro) and
0.06% for the yen. Emerging market currencies have higher spreads: for
instance, s ¼ 0.12% for Korea and 0.10% for Brazil. During the same
period, the volatility of the exchange rate was s ¼ 10% for the German
mark, 12% for the Japanese yen, 17% for Brazil and 18% for Korea. In
Panel B of Table 1, we compute D with s ¼ 10%, a realistic value for the
euro and yen. As we noted above, if the sole source of the noise were a bid/
ask spread of size s, then a should be set to s/2. Therefore, Panel B reports
the values of D for values of a ranging from 0.02% to 0.1%. For example,
the dollar/euro or dollar/yen exchange rates (calibrated to s ¼ 10%,
a ¼ 0.02%) should be sampled every D ¼ 23 min if the overall sample
length is T ¼ 1 day, and every 1.1 h if T ¼ 1 year.
Furthermore, using the bid/ask spread alone as a proxy for all microstructure frictions will lead, except in unusual circumstances, to an understatement of the parameter a, since variances are additive. Thus, since D is
increasing in a, one should interpret the value of D read off 1 on the row
corresponding to a ¼ s/2 as a lower bound for the optimal sampling
interval.
4.3 Monte Carlo Evidence
To validate empirically these results, we perform Monte Carlo simulations. We simulate M ¼ 10,000 samples of length T ¼ 1 year of the process
~ , and then
X, add microstructure noise U to generate the observations X
366
Sampling and Market Microstructure Noise
the log returns Y. We sample the log-returns at various intervals D ranging
from 5 min to 1 week, and calculate the bias and variance of the estimator
s
^ 2 over the M simulated paths. We then compare the results to the
theoretical values given in Equations (17) and (18) of Theorem 1. The
noise distribution is Gaussian, s ¼ 30% and a ¼ 0.15% — the values
we calibrated to stock returns data above. Table 2 shows that the theoretical values are in close agreement with the results of the Monte Carlo
simulations.
Table 2 also illustrates the magnitude of the bias inherent in sampling
at too high a frequency. While the value of s2 used to generate the data is
0.09, the expected value of the estimator when sampling every 5 min is
0.18, so on average the estimated quadratic variation is twice as big as it
should be in this case.
5. Incorporating Market Microstructure Noise Explicitly
So far we have stuck to the sum of squares of log-returns as our estimator
of volatility. We then showed that, for this estimator, the optimal sampling frequency is finite. However, this implies that one is discarding a
large proportion of the high frequency sample (299 out of every 300
observations in the example described in the introduction), in order to
mitigate the bias induced by market microstructure noise. Next, we show
that if we explicitly incorporate the Us into the likelihood function, then
we are back in a situation where the optimal sampling scheme consists in
sampling as often as possible — that is, using all the data available.
Specifying the likelihood function of the log-returns, while recognizing
that they incorporate noise, requires that we take a stand on the distribution of the noise term. Suppose for now that the microstructure noise is
normally distributed, an assumption whose effect we will investigate
below in Section 6. Under this assumption, the likelihood function for
the Ys is given by
1
l h, g2 ¼ ln detðV Þ=2 N ln 2pg 2 =2 2g 2 Y 0 V 1 Y , ð25Þ
where the covariance matrix for the vector Y ¼ (Y1, . . . ,YN)0 is given by
g 2V, where
1
0
1 þ h2
h
0
0
B
.. C
B h
1 þ h2
h
»
. C
C
B
C
B
B
2
V ¼ vij i; j¼1; ... ;N ¼ B 0
h
1þh
»
0 C
C ð26Þ
C
B
C
B ..
@ .
»
»
»
h A
0
0
h 1 þ h2
367
The Review of Financial Studies / v 18 n 2 2005
Further,
detðV Þ ¼
1 h2Nþ2
1 h2
ð27Þ
and, neglecting the end effects, an approximate inverse of V is the matrix
V ¼ [vij]i, j¼1, . . . , N where
1
vij ¼ 1 h2 ðhÞjijj
[see Durbin (1959)]. The product VV differs from the identity matrix only
on the first and last rows. The exact inverse is V1 ¼ [vij]i, j ¼ 1, . . . ,N where
1 1 n
vij ¼ 1 h2
1 h2Nþ2
ðhÞjijj ðhÞiþj ðhÞ2Nijþ2
o
ðhÞ2Nþjijjþ2 þ ðhÞ2Nþijþ2 þ ðhÞ2Niþjþ2
ð28Þ
[see Shaman (1969), Haddad (1995)].
From the perspective of practical implementation, this estimator is
nothing else than the MLE estimator of an MA(1) process with Gaussian
errors: any existing computer routines for the MA(1) situation can, therefore, be applied [see e.g., Hamilton (1995, Section 5.4)]. In particular, the
likelihood function can be expressed in a computationally efficient form
by triangularizing the matrix V, yielding the equivalent expression:
N
N ~2
Yi
1X
1X
l h, g2 ¼ lnð2pdi Þ ,
2 i¼1
2 i¼1 di
ð29Þ
where
di ¼ g 2
1 þ h2 þ þ h2i
1 þ h2 þ þ h2ði 1Þ
~ i s are obtained recursively as Y
~ 1 ¼ Y1 and for i ¼ 2, . . . , N:
and the Y
h 1 þ h2 þ þ h2ði 2Þ ~
~
Yi ¼ Yi Yi1 :
1 þ h2 þ þ h2ði 1Þ
This latter form of the log-likelihood function involves only single sums as
opposed to double sums if one were to compute Y 0 V 1Y by brute force
using the expression of V 1 given above.
We now compute the distribution of the MLE estimators of s2 and a2,
which follows by the delta method from the classical result for the MA(1)
estimators of g and h by the following proposition.
368
Sampling and Market Microstructure Noise
Proposition 1. When U is normally distributed, the MLE ð^
s2 , ^a2 Þ is
consistent and its asymptotic variance is given by
2 2
^ ,^
a
avarnormal s
0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
6
4 s Dð4a2 þ s2 DÞ þ 2s4 D
s2 Dh D, s2 , a2
¼@
A,
D 2
2a þ s2 D h D, s2 , a2
2
with
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h D, s2 , a2 2a2 þ s2 Dð4a2 þ s2 DÞ þ s2 D:
ð30Þ
Since avarnormal ð^
s2 Þ is increasing in D, it is optimal to sample as often as
possible. Further, since
2
avarnormal s
^ ¼ 8s3 aD1=2 þ 2s4 D þ oðDÞ,
ð31Þ
the loss of efficiency relative to the case where no market microstructure
noise is present (and, if a2¼0 is not estimated, avarð^
s2 Þ ¼ 2s4 D as given
2
in Equation (7), or if a ¼0 is estimated, avar(s)¼6^
s4 d) is at order D1/2.
2
Figure 3 plots the asymptotic variances of s
^ as functions of D with and
without noise (the parameter values are again s ¼ 30% and a ¼ 0.15%).
Figure 4 reports histograms of the distributions of s
^ 2 and ^a2 from 10,000
Monte Carlo simulations with the solid curve plotting the asymptotic
distribution of the estimator from Proposition 1. The sample path is of
length T ¼ 1 year, the parameter values are the same as above, and the
Figure 3
^ 2 without and with noise taken into account
Comparison of the asymptotic variances of the MLE s
369
The Review of Financial Studies / v 18 n 2 2005
Figure 4
^ 2 , a^2 ) with Gaussian microstructure noise
Asymptotic and Monte Carlo distributions of the MLE (s
process is sampled every 5 min — since we are now accounting explicitly
for the presence of noise, there is no longer a reason to sample at lower
frequencies. Indeed, the figure documents the absence of bias and the
good agreement of the asymptotic distribution with the small sample one.
6. The Effect of Misspecifying the Distribution of the
Microstructure Noise
We now study a situation where one attempts to incorporate the presence
of the Us into the analysis, as in Section 5, but mistakenly assumes a
370
Sampling and Market Microstructure Noise
misspecified model for them. Specifically, we consider the case where the
Us are assumed to be normally distributed when in reality they have a
different distribution. We still suppose that the Us are i.i.d. with mean zero
and variance a2.
Since the econometrician assumes the Us to have a normal distribution,
inference is still done with the log-likelihood l(s2, a2), or equivalently
l(h, g 2) given in Equation (25), using Equations (9) and (10). This means
that the scores l_s2 and l_a2 , or equivalently Equations (C.1) and (C.2) are
used as moment functions (or ‘‘estimating equations’’). Since the first
order moments of the moment functions only depend on the second
order moment structure of the log-returns (Y1, . . . ,YN), which is
unchanged by the absence of normality, the moment functions are unbiased under the true distribution of the Us:
h i
h i
Etrue l_h ¼ Etrue l_g2 ¼ 0
ð32Þ
s2 , ^a2 Þ based on these
and similarly for l_s2 and l_a2 . Hence the estimator ð^
moment functions is consistent and asymptotically unbiased (even though
the likelihood function is misspecified).
The effect of misspecification, therefore, lies in the asymptotic variance
matrix. By using the cumulants of the distribution of U, we express the
asymptotic variance of these estimators in terms of deviations from normality. But as far as computing the actual estimator, nothing has changed
relative to Section 5: we are still calculating the MLE for an MA(1)
process with Gaussian errors and can apply exactly the same computational routine.
However, since the error distribution is potentially misspecified, one
could expect the asymptotic distribution of the estimator to be altered.
This does not happen, as far as s
^ 2 is concerned: see the following theorem.
^2 Þ obtained by maximizing the possibly
Theorem 2. The estimators ð^
s2 , a
misspecified log-likelihood function (25) are consistent and their asymptotic
variance is given by
2 2
2 2
0 0
^ ,^
a ¼ avarnormal s
^ , ^a þ cum4 ½U ,
ð33Þ
avartrue s
0 D
^2 Þ is the asymptotic variance in the case where the
where avarnormal ð^
s2 , a
distribution of U is normal, that is, the expression given in Proposition 1.
In other words, the asymptotic variance of s
^ 2 is identical to its expression
if the Us had been normal. Therefore, the correction we proposed for
the presence of market microstructure noise relying on the assumption
that the noise is Gaussian is robust to misspecification of the error
distribution.
371
The Review of Financial Studies / v 18 n 2 2005
Documenting the presence of the correction term through simulations
presents a challenge. At the parameter values calibrated to be realistic, the
order of magnitude of a is a few basis points, say a ¼ 0.10% ¼ 103. But if
U if of order 103, cum4[U] which is of the same order as U 4, is of order
1012. In other words, with a typical noise distribution, the correction
term in Equation (33) will not be visible.
Nevertheless, to make it discernible, we use a distribution for U with the
same calibrated standard deviation a as before, but a disproportionately
large fourth cumulant. Such a distribution can be constructed by letting
U ¼ vTn where v > 0 is constant and Tn is a Student t distribution with v
degrees of freedom. Tn has mean zero, finite variance as long as v > 2 and
finite fourth moment (hence finite fourth cumulant) as long as v > 4. But
as v approaches 4 from above, E½Tn4 tends to infinity. This allows us to
produce an arbitrarily high value of cum4[U] while controlling for the
magnitude of the variance. The specific expressions of a2 and cum4[U] for
this choice of U are given by
a2 ¼ var½U ¼
cum4 ½U ¼
v2 n
,
n2
6v4 n2
ðn 4Þðn 2Þ2
:
ð34Þ
ð35Þ
Thus, we can select the two parameters (v, n) to produce desired values of
(a2, cum4[U ]). As before, we set a ¼ 0.15%. Then, given the form of the
asymptotic variance matrix Equation (33), we set cum4[U ] so that
cum4 ½UD ¼ avarnormal ð^
a2 Þ=2. This makes avartrue ð^a2 Þ by construction
50% larger than avarnormal ð^
a2 Þ. The resulting values of (v, n) from solving
Equations (34) and (35) are v ¼ 0.00115 and v ¼ 4.854. As above, we set
the other parameters to s ¼ 30%, T ¼ 1 year, and D ¼ 5 minutes. Figure 5
reports histograms of the distributions of s
^ 2 and ^a2 from 10,000 Monte
Carlo simulations. The solid curve plots the asymptotic distribution of the
estimator, given now by Equation (33). There is again good adequacy
between the asymptotic and small sample distributions. In particular, we
note that as predicted by Theorem 2, the asymptotic variance of s
^ 2 is
2
unchanged relative to Figure 4 while that of ^a is 50% larger. The small
sample distribution of s
^ 2 appears unaffected by the non-Gaussianity of
the noise; with a skewness of 0.07 and a kurtosis of 2.95, it is closely
approximated by its asymptotic Gaussian limit. The small sample distribution of ^
a2 does exhibit some kurtosis (4.83), although not large relative
to that of the underlying noise distribution (the values of v and n imply a
kurtosis for U of 3 þ 6/(n 4) ¼ 10). Similar simulations but with a longer
time span of T ¼ 5 years are even closer to the Gaussian asymptotic limit:
the kurtosis of the small sample distribution of ^a2 goes down to 2.99.
372
Sampling and Market Microstructure Noise
Figure 5
^ 2 , a^2 ) with misspecified microstructure noise
Asymptotic and Monte Carlo distributions of the QMLE (s
7. Robustness to Misspecification of the Noise Distribution
Going back to the theoretical aspects, the above Theorem 2 has implications for the use of the Gaussian likelihood l that go beyond consistency,
namely that this likelihood can also be used to estimate the distribution of
s
^ 2 under misspecification. With l denoting the log-likelihood assuming
that the Us are Gaussian, given in Equation (25), €l ð^
s2 , ^a2 Þ denote the
observed information matrix in the original parameters s2 and a2. Then
1 € 2 2 1
^
d normal ¼ l s
^ , ^a
V ¼ avar
T
373
The Review of Financial Studies / v 18 n 2 2005
is the usual estimate of asymptotic variance when the distribution is
correctly specified as Gaussian. Also note, however, that otherwise, so
^ is also a consistent estimate of the matrix
long as ð^
s2 , ^
a2 Þ is consistent, V
2 ^2
avarnormal ð^
s , a Þ. Since this matrix coincides with avartrue ð^
s2 , ^a2 Þ for all
2
2
but the (a , a ) term (see Equation (33)), the asymptotic variance of
^ s2 s2 . The similar statement is
T 1=2 ð^
s2 s2 Þ is consistently estimated by V
true for the covariances, but not, obviously, for the asymptotic variance of
T 1=2 ð^
a2 a2 Þ.
In the likelihood context, the possibility of estimating the asymptotic
variance by the observed information is due to the second Bartlett identity. For a general log likelihood l, if S Etrue ½l_l_0 =N and D Etrue ½€l =N
(differentiation refers to the original parameters (s2, a2), not the transformed parameters (g 2, h)) this identity says that
S D ¼ 0:
ð36Þ
It implies that the asymptotic variance takes the form
1
avar ¼ D DS 1 D ¼ DD1 :
ð37Þ
It is clear that Equation (37) remains valid if the second Bartlett identity
holds only to first order, that is,
S D ¼ oð1Þ
ð38Þ
as N ! 1, for a general criterion function l which satisfies Etrue ½l_ ¼ oðNÞ.
However, in view of Theorem 2, Equation (38) cannot be satisfied. In
fact, we show in Appendix E that
S D ¼ cum4 ½U gg0 þ oð1Þ,
ð39Þ
where
g¼
gs 2
ga2
0
D1=2
3=2
þ s 2 DÞ
1
B
C
C:
¼B
@ 1
D1=2 sð6a2 þ s2 DÞ A
1
3=2
ð4a2 þ s2 DÞ
2a4
sð4a2
ð40Þ
From Equation (40), we see that g 6¼ 0 whenever s2 > 0. This is consistent
with the result in Theorem 2 that the true asymptotic variance matrix,
avartrue ð^
s2 , ^
a2 Þ; does not coincide with the one for Gaussian noise,
avarnormal ð^
s2 , ^
a2 Þ. On the other hand, the 2 2 matrix gg0 is of rank 1,
signaling that there exist linear combinations that will cancel out the first
column of S D. From what we already know of the form of the correction matrix, D1 gives such a combination that the asymptotic variance of
the original parameters (s2, a2) will have the property that its first column
is not subject to correction in the absence of normality.
374
Sampling and Market Microstructure Noise
A curious consequence of Equation (39) is that while the observed
information can be used to estimate the asymptotic variance of s
^ 2 when
2
2
a is not known, this is not the case when a is known. This is because the
second Bartlett identity also fails to first order when considering a2 to
be known, that is, when differentiating with respect to s2 only. Indeed,
in that case we have from the upper left component in the matrix
Equation (39):
h
h
2 i
i
Ss2 s2 Ds2 s2 ¼ N 1 Etrue is2 s2 s2 , a2
þ N 1 Etrue €ls2 s2 s2 , a2
¼ cum4 ½U Þðgs2 Þ2 þ oð1Þ,
which is not o(1) unless cum4 [U ] ¼ 0.
To make the connection between Theorem 2 and the second Bartlett
identity, one needs to go to the log profile likelihood
l s2 sup l s2 , a2 :
ð41Þ
a2
Obviously, maximizing the likelihood l(s2, a2) is the same as maximizing
l(s2). Thus one can think of s2 as being estimated (when a2 is unknown)
by maximizing the criterion function l(s2), or by solving l_ ð^
s2 Þ ¼ 0. Also,
the observed profile information is related to the original observed
information by
2 1 h 2 2 1 i
€ s
l
^
¼ €l s
^ , ^a
s2 s2
,
ð42Þ
that is, the first (upper left hand corner) component of the inverse
observed information in the original problem. We explain this in
Appendix E, where we also show that Etrue ½l_ ¼ oðNÞ. In view of Theorem
€ð^
2, l
s2 Þ can be used to estimate the asymptotic variance of s
^ 2 under the
true (possibly non-Gaussian) distribution of the Us, and so it must be that
the criterion function l satisfies Equation (38), that is
h 2 i
2 € s
N 1 Etrue l_ s2
¼ oð1Þ:
þ N 1 Etrue l
ð43Þ
This is indeed the case, as shown in Appendix E.
This phenomenon is related, although not identical, to what occurs in
the context of quasi-likelihood [for comprehensive treatments of quasilikelihood theory, see the books by McCullagh and Nelder (1989) and
Heyde (1997), and the references therein, and for early econometrics
examples, see Macurdy (1982) and White (1982)]. In quasi-likelihood
situations, one uses a possibly incorrectly specified score vector which is
375
The Review of Financial Studies / v 18 n 2 2005
nevertheless required to satisfy the second Bartlett identity. What makes
our situation unusual relative to quasi-likelihood is that the interest
parameter s2 and the nuisance parameter a2 are entangled in the same
estimating equations (l_s2 and l_a2 from the Gaussian likelihood) in such a
way that the estimate of s2 depends, to first order, on whether a2 is known
or not. This is unlike the typical development of quasi-likelihood, where
the nuisance parameter separates out [see, e.g., McCullagh and Nelder
(1989, Table 9.1, p. 326)]. Thus only by going to the profile likelihood l
can one make the usual comparison to quasi-likelihood.
8. Randomly Spaced Sampling Intervals
One essential feature of transaction data in finance is that the time that
separates successive observations is random, or at least time-varying. So,
as in Aı̈t-Sahalia and Mykland (2003), we are led to consider the case
where Di ¼ t i t i1 are either deterministic and time-varying, or random
in which case we assume for simplicity that they are i.i.d., independent of
the W process. This assumption, while not completely realistic [see Engle
and Russell (1998) for a discrete time analysis of the autoregressive
dependence of the times between trades] allows us to make explicit calculations at the interface between the continuous and discrete time scales.
We denote by NT the number of observations recorded by time T. NT is
random if the Ds are. We also suppose that Uti can be written Ui, where the
Ui are i.i.d. and independent of the W process and the Dis. Thus, the
observation noise is the same at all observation times, whether random or
nonrandom. If we define the Yis as before, in the first two lines of
Equation (8), though the MA(1) representation is not valid in the same
form.
We can do inference conditionally on the observed sampling times, in
light of the fact that the likelihood function using all the available information is
LðYN , DN , . . . , Y1 , D1 ; b, cÞ ¼ LðYN , . . . , Y1 jDN , . . . , D1 ; bÞ
LðDN , . . . , D1 ; cÞ,
where b are the parameters of the state process, that is (s2, a2), and c are
the parameters of the sampling process, if any (the density of the sampling
intervals density L(DNT, . . . , D1; c) may have its own nuisance parameters
c, such as an unknown arrival rate, but we assume that it does not depend
on the parameters b of the state process). The corresponding loglikelihood function is
N
X
n¼1
376
ln LðYN , . . . , Y1 jDN , . . . , D1 ; bÞ þ
N
1
X
n¼1
ln LðDN , . . . , D1 ; cÞ
ð44Þ
Sampling and Market Microstructure Noise
and since we only care about b, we only need to maximize the first term in
that sum.
We operate on the covariance matrix of the log-returns Ys, now
given by
1
0
s2 D1 þ 2a2
a2
0
0
C
B
..
C
B
C
B
s2 D2 þ 2a2
a2
»
.
a2
C
B
C
B
2
2
2
¼B
C: ð45Þ
0
a
s D3 þ 2a
»
0
C
B
C
B
.
..
2
C
B
»
»
»
a
A
@
0
a2
0
s2 Dn þ 2a2
Note that in the equally spaced case, ¼ g2V. But now Y no longer
follows an MA(1) process in general. Furthermore, the time variation in
Dis gives rise to heteroskedasticity as is clear from the diagonal elements of
. This is consistent with the predictions of the model of Easley and
~ is
O’Hara (1992) where the variance of the transaction price process X
heteroskedastic as a result of the influence of the sampling times. In their
model, the sampling times are autocorrelated and correlated with the
evolution of the price process, factors we have assumed away here.
However, Aı̈t-Sahalia and Mykland (2003) show how to conduct likelihood inference in such a situation.
The log-likelihood function is given by
ln LðYN , . . . , Y1 jDN , . . . , D1 ; bÞ
l s2 , a2 ¼ ln detðÞ=2 Nlnð2pÞ=2 Y 0 1 Y =2:
ð46Þ
In order to calculate this log-likelihood function in a computationally
efficient manner, it is desirable to avoid the ‘‘brute force’’ inversion of
the N N matrix . We extend the method used in the MA(1) case (see
Equation (29)) as follows. By Theorem 5.3.1 in Dahlquist and Bj€
orck
(1974), and the development in the proof of their Theorem 5.4.3, we can
decompose in the form ¼ LDLT, where L is a lower triangular matrix
whose diagonals are all 1 and D is diagonal. To compute the relevant quantities, their Example 5.4.3 shows that if one writes D ¼
diag(g1, . . . , gn) and
1
0
1
0
0 0
.. C
B
B k2
1
0
»
.C
C
B
B0 k
1 » 0C
ð47Þ
L¼B
C,
3
C
B
.
C
B .
@ .
» » » 0A
0
0
kn
1
377
The Review of Financial Studies / v 18 n 2 2005
then the gk s and kk s follow the recursion equation g1 ¼ s2D1 þ 2a2 and for
i ¼ 2, . . . , N:
ki ¼ a2 =gi1
and
gi ¼ s2 Di þ 2a2 þ ki a2 :
ð48Þ
~ ¼ L1 Y so that Y 0 1 Y ¼ Y
~ . From Y ¼ LY
~ , it
~ 0 D1 Y
Then, define Y
~
follows that Y1 ¼ Y1 and, for i ¼ 2, . . . , N:
~ i ¼ Yi ki Y
~ i1 :
Y
And det() ¼ det(D) since det(L) ¼ 1. Thus we have obtained a computationally simple form for Equation (46) that generalizes the MA(1) form in
Equation (29) to the case of non-identical sampling intervals:
N
N ~2
Yi
1X
1X
l s 2 , a2 ¼ lnð2pgi Þ :
2 i¼1
2 i¼1 gi
ð49Þ
We can now turn to statistical inference using this likelihood function.
s2 s2 , ^a2 a2 Þ is of the form
As usual, the asymptotic variance of T 1=2 ð^
01 h
i 1 h
i 11
E €ls2 s2
E €ls2 a2
2 2
B
C
T
avar s
^ ,^
a ¼ lim @ T
ð50Þ
h
iA :
1
T !1
€
E la2 a2
T
To compute this quantity, suppose in the following that b1 and b2 can
represent either s2 or a2. We start with:
Lemma 2. Fisher’s Conditional Information is given by
h
i
1 q2 ln det E €lb2 b1 D ¼ :
2 qb2 b1
ð51Þ
To compute the asymptotic distribution of the MLE of (b1, b2), one would
then need to compute the inverse of E½€lb2 b1 ¼ ED ½E½€lb2 b1 jD where ED
denotes expectation taken over the law of the sampling intervals. From
Equation (51), and since the order of ED and q2/qb2b1 can be interchanged, this requires the computation of
ED ½ln det ¼ ED ½ln det D ¼
N
X
ED ½lnðgi Þ,
i¼1
where from Equation (48) the gis are given by the continuous fraction
g1 ¼ s2 D1 þ 2a2 ,
g3 ¼ s2 D3 þ 2a2 g2 ¼ s2 D2 þ 2a2 a4
s2 D2 þ 2a2 378
a4
,
s2 D1 þ 2a2
a4
s2 D1 þ 2a2
Sampling and Market Microstructure Noise
and in general
a4
gi ¼ s2 Di þ 2a2 s2 Di1 þ 2a2 a4
»
It, therefore, appears that computing the expected value of ln(gi) over the
law of (D1, D2, . . . , Di) will be impractical.
8.1 Expansion around a fixed value of D
To continue further with the calculations, we propose to expand around a
fixed value of D, namely D0 ¼ E [D]. Specifically, suppose now that
Di ¼ D0 ð1 þ eji Þ,
ð52Þ
where e and D0 are nonrandom, the jis are i.i.d. random variables with
mean zero and finite distribution. We will Taylor-expand the expressions
above around e ¼ 0, that is, around the non-random sampling case we
have just finished dealing with. Our expansion is one that is valid when the
randomness of the sampling intervals remains small, that is, when var[Di]
is small, or o(1). Then we have D0 ¼ E [D] ¼ O(1) and var½Di ¼ D20 e2 var½ji .
The natural scaling is to make the distribution of ji finite, that is,
var[ji] ¼ O(1), so that e2 ¼ O(var[Di]) ¼ o(1). But any other choice would
have no impact on the result since var[Di] ¼ o(1) implies that the product
e2var[ji] is o(1) and whenever we write remainder terms below they can be
expressed as Op(e3j3) instead of just O(e3). We keep the latter notation for
clarity given that we set ji ¼ Op(1). Furthermore, for simplicity, we take
the jis to be bounded.
We emphasize that the time increments or durations Di do not tend to
zero length as e ! 0. It is only the variability of the Dis that goes to zero.
Denote by 0 the value of when D is replaced by D0, and let X denote
the matrix whose diagonal elements are the terms D0ji, and whose
off-diagonal elements are zero. We obtain the following theorem.
Theorem 3. The MLE ð^
s2 , ^
a2 Þ is again consistent, this time with asymptotic
variance
2 2
ð53Þ
avar s
^ ,^
a ¼ Að0Þ þ e2 Að2Þ þ O e3 ,
where
Að0Þ
0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 s6 D0 ð4a2 þ s2 D0 Þ þ 2s4 D0
@
¼
1
s2 D0 h D0 , s2 , a2
A
D0 2
2a þ s2 D0 h D0 , s2 , a2
2
379
The Review of Financial Studies / v 18 n 2 2005
and
ð2Þ
A
var½j
¼
2
ð4a þ D0 s2 Þ
As2 s2
ð2Þ
As2 a2
ð2Þ
Aa2 a2
!
ð2Þ
with
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2Þ
3=2
As2 s2 ¼ 4ðD20 s6 þ D0 s5 4a2 þ D0 s2 Þ,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2Þ
3=2
As2 a2 ¼ D0 s3 4a2 þ D0 s2 ð2a2 þ 3D0 s2 Þ þ D20 s4 ð8a2 þ 3D0 s2 Þ,
pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2Þ
Aa2 a2 ¼ D20 s2 ð2a2 þ s D0 4a2 þ D0 s2 þ D0 s2 Þ2 :
In connection with the preceding result, we underline that the quantity
avarð^
s2 , ^
a2 Þ is a limit as T ! 1, as in Equation (50). Equation (53),
therefore, is an expansion in e after T ! 1.
Note that A(0) is the asymptotic variance matrix already present in
Proposition 1, except that it is evaluated at D0 = E[D]. Note also that the
second order correction term is proportional to var[j], and is therefore
zero in the absence of sampling randomness. When that happens, D ¼ D0
with probability one and the asymptotic variance of the estimator reduces
to the leading term A(0), that is, to the result in the fixed sampling case
given in Proposition 1.
8.2 Randomly spaced sampling intervals and misspecified
microstructure noise
Suppose now, as in Section 6, that the Us are i.i.d., have mean zero and
variance a2, but are otherwise not necessarily Gaussian. We adopt the
same approach as in Section 6, namely to express the estimator’s properties in terms of deviations from the deterministic and Gaussian case. The
additional correction terms in the asymptotic variance are given in the
following result.
Theorem 4. The asymptotic variance is given by
avartrue s
^, ^
a2 ¼ Að0Þ þ cum4 ½U Bð0Þ þ e2 Að2Þ þ cum4 ½U Bð2Þ þ O e3
ð54Þ
where A(0) and A(2) are given in the statement of Theorem 3 and
B
380
ð0Þ
¼
0
0
0
D0
Sampling and Market Microstructure Noise
while
B
ð2Þ
¼ var½j
3=2
ð2Þ
Bs 2 s 2
ð2Þ
Bs2 a2 ¼
ð2Þ
Ba2 a2
¼
10D0 s5
ð4a2 þ D0 s2 Þ
5=2
þ
Bs2 s2
ð 2Þ
Bs2 a2
ð2Þ
Ba2 a2
!
ð 2Þ
4D20 s6 16a4 þ 11a2 D0 s2 þ 2D20 s4
3
ð2a2 þ D0 s2 Þ ð4a2 þ D0 s2 Þ
2
D20 s4
3
5=2
ð2a2 þ D0 s2 Þ ð4a2 þ D0 s2 Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4a2 þ D0 s2 32a6 þ 64a4 D0 s2 þ 35a2 D20 s4 þ 6D30 s6
1=2 þ D0 s 116a6 þ 126a4 D0 s2 þ 47a2 D20 s4 þ 6D30 s6
5=2
16a8 D0 s3 13a4 þ 10a2 D0 s2 þ 2D20 s4
¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 :
3
5=2
ð2a2 þ D0 s2 Þ ð4a2 þ D0 s2 Þ
2a2 þ s2 D s2 Dð4a2 þ s2 DÞ
The term A(0) is the base asymptotic variance of the estimator, already
present with fixed sampling and Gaussian noise. The term cum4[U ]B(0) is
the correction due to the misspecification of the error distribution. These
two terms are identical to those present in Theorem 2. The terms proportional to e2 are the further correction terms introduced by the randomness
of the sampling. A(2) is the base correction term present even with
Gaussian noise in Theorem 3, and cum4 [U ]B(2) is the further correction
due to the sampling randomness. Both A(2) and B(2) are proportionalto
var[j] and hence vanish in the absence of sampling randomness.
9. Extensions
In this section, we briefly sketch four extensions of our basic model. First,
we show that the introduction of a drift term does not alter our conclusions. Then we examine the situation where market microstructure noise is
serially correlated; there, we show that the insight of Theorem 1 remains
valid, namely that the optimal sampling frequency is finite. Third, we turn
to the case where the noise is correlated with the efficient price signal.
Fourth, we discuss what happens if volatility is stochastic.
In a nutshell, each one of these assumptions can be relaxed without
affecting our main conclusion, namely that the presence of the noise gives
rise to a finite optimal sampling frequency. The second part of our analysis, dealing with likelihood corrections for microstructure noise, will not
381
The Review of Financial Studies / v 18 n 2 2005
necessarily carry through unchanged if the assumptions are relaxed (for
instance, there is not even a known likelihood function if volatility is
stochastic, and the likelihood must be modified if the assumed variancecovariance structure of the noise is modified).
9.1 Presence of a drift coefficient
What happens to our conclusions when the underlying X process has a
drift? We shall see in this case that the presence of the drift does not alter
our earlier conclusions. As a simple example, consider linear drift, that is,
replace Equation (2) with
Xt ¼ mt þ sWt :
ð55Þ
The contamination by market microstructure noise is as before: the
observed process is given by Equation (3).
~ ti X
~ ti1 þ
As before, we first-difference to get the log-returns Yi ¼ X
Uti Uti1 . The likelihood function is now
lnLðY1 , ..., YN jDN , .. .,D1 ; bÞ
l s2 ,a2 ,m ¼ ln detðÞ=2 N lnð2pÞ=2 ðY mDÞ0 1 ðY mDÞ=2,
where the covariance matrix is given in Equation (45), and where
D ¼ (D1, . . . , DN)0 . If b denotes either s2 or a2, one obtains
1
€lmb ¼ D0 q ðY mDÞ,
qb
so that E½€lmb jD ¼ 0 no matter whether the Us are normally distributed or
have another distribution with mean 0 and variance a2. In particular,
h i
E €lmb ¼ 0:
ð56Þ
Now let E½€l be the 3 3 matrix of expected second likelihood derivatives.
Let E½€l ¼ TE½DD þ oðTÞ. Similarly define covðl_, l_Þ ¼ TE½DS þ oðTÞ.
As before, when the Us have a normal distribution, S ¼ D, and otherwise
that is not the case. The asymptotic variance matrix of the estimators is of
the form avar ¼ E [D]D1SD1.
Let Ds2;a2 be the corresponding 2 2 matrix when estimation is carried
out on s2 and a2 for known m, and Dm is the asymptotic information on m
for known s2 and a2. Similarly define Ss2;a2 and avars2;a2. Since D is block
diagonal by Equation (56),
D s 2 ; a2 0
D¼
,
00
Dm
382
Sampling and Market Microstructure Noise
it follows that
D
Hence
1
¼
D1
s2 ;a2
0
00
D1
m
!
:
2 2
1
avar s
^ ,^
a ¼ E ½DD1
s2 ;a2 Ss2 ;a2 Ds2 ;a2 :
ð57Þ
The asymptotic variance ofð^
s2 , ^
a2 Þ is thus the same as if m were known, in
other words, as if m ¼ 0, which is the case that we focused on in all the
previous sections.
9.2 Serially correlated noise
We now examine what happens if we relax the assumption that the market
microstructure noise is serially independent. Suppose that, instead of being
i.i.d. with mean 0 and variance a2, the market microstructure noise follows
dUt ¼ bUt dt þ cdZt ,
ð58Þ
where b > 0, c > 0 and Z is a Brownian motion independent of W. UDjU0
has a Gaussian distribution with mean ebDU0 and variance c2/2b(1 e2bD). The unconditional mean and variance of U are 0 and a2 = c2/2b.
The main consequence of this model is that the variance contributed by
the noise to a log-return observed over an interval of time D is now of
order O(D), that is of the same order as the variance of the efficient price
process s2D, instead of being of order O(1) as previously. In other words,
log-prices observed close together have very highly correlated noise terms.
Because of this feature, this model for the microstructure noise would be
less appropriate if the primary source of the noise consists of bid-ask
bounces. In such a situation, the fact that a transaction is on the bid or
ask side has little predictive power for the next transaction, or at least not
enough to predict that two successive transactions are on the same side
with very high probability [although Choi, Salandro, and Shastri (1988)
have argued that serial correlation in the transaction type can be a component of the bid-ask spread, and extended the model of Roll (1984) to
allow for it]. On the other hand, the model (58) can better capture effects
such as the gradual adjustment of prices in response to a shock such as a
large trade. In practice, the noise term probably encompasses both of
these examples, resulting in a situation where the variance contributed
by the noise has both types of components, some of order O(1), some of
lower orders in D.
The observed log-returns take the form
~ ti X
~ ti1 þ Uti Uti1
Yi ¼ X
¼ sðWti Wti1 Þ þ Uti Uti1
wi þ ui ,
383
The Review of Financial Studies / v 18 n 2 2005
where the wis are i.i.d. N(0, s2D), the uis are independent of the wis, so we
have var½Yi ¼ s2 D þ E½u2i , and they are Gaussian with mean zero and
variance
h
i c2 1 ebD 2
2
¼ c2 D þ oðDÞ
E ui ¼ E ðUti Uti1 Þ ¼
ð59Þ
b
instead of 2a2.
In addition, the uis are now serially correlated at all lags since
c2 1 ebDðikÞ
E ½Uti Utk ¼
2b
for i k. The first-order correlation of the log-returns is now
2
c2 1 ebD
c2 b 2
D þ o D2
covðYi , Yi1 Þ ¼ ¼
2
2b
instead of h.
The result analogous to Theorem 1 is as follows. If one ignores the
presence of this type of serially correlated noise when estimating s2, then
follows the theorem.
Theorem 5. In small samples (finite T), the RMSE of the estimator s
^ 2 is
given by
2Tb
4
bD 2 T 2bD
e
c
1
e
1
þ
e
2
2
c4 1 ebD
D
RMSE s
^ ¼
þ
:
2
2
2
2
2
b D
T b ð1 þ ebD Þ
2 !1=2
c2 1 ebD
2
2
s Dþ
þ
TD
b
2
2
2
s þ c2 D
bc2
1
2
Dþ
þO D þO 2
¼ c c2 T
T
2
ð60Þ
so that for large T, starting from a value of c2 in the limit where D ! 0,
increasing D first reduces RMSE [^
s2 ]. Hence the optimal sampling
frequency is finite.
One would expect this type of noise to be not nearly as bad as i.i.d. noise
for the purpose of inferring s2 from high frequency data. Indeed, the
variance of the noise is of the same order O(D) as the variance of the
efficient price process. Thus log returns computed from transaction prices
sampled close together are not subject to as much noise as previously
384
Sampling and Market Microstructure Noise
(O(D) versus O(1)) and the squared bias b2 of the estimator s
^ 2 no longer
4
diverges to infinity as D ! 0: it has the finite limit c . Nevertheless, b2 first
decreases as D increases from 0, since
2
2
c4 1 ebD
2
2 2
b ¼ E s
^ s ¼
b2 D2 e2bD
and qb2/qD ! bc4 < 0 as D ! 0. For large enough T, this is sufficient to
generate a finite optimal sampling frequency.
To calibrate the parameter values b and c, we refer to the same empirical
microstructure studies we mentioned in Section 4. We now have
p ¼ E½u2i =ðs2 D þ E½u2i Þ as the proportion of total variance that is
microstructure-induced; we match it to the numbers in Equation (24)
from Madhavan, Richardson, and Roomans (1997). In their Table 5,
they report the first-order correlation of price changes (hence returns) to
be approximately r ¼ 0.2 at their frequency of observation. Here
r ¼ cov(Yi, Yi1)/var[Yi]. If we match p ¼ 0.6 and r ¼ 0.2, with
s ¼ 30% as before, we obtain (after rounding) c ¼ 0.5 and b ¼ 3 104.
Figure 6 displays the resulting RMSE of the estimator as a function of D
and T. The overall picture is comparable to Figure 2.
As for the rest of the analysis of the article, dealing with likelihood
corrections for microstructure noise, the covariance matrix of the logreturns, g2V in Equation (26) should be replaced by the matrix whose
diagonal elements are
c2 1 ebD
var Yi2 ¼ E w2i þ E u2i ¼ s2 D þ
b
Figure 6
^ 2 when the presence of serially correlated noise is ignored
RMSE of the estimator s
385
The Review of Financial Studies / v 18 n 2 2005
and off-diagonal elements i > j are:
covðYi , Yj Þ ¼ E Yi Yj ¼ E ðwi þ ui Þðwj þ uj Þ
¼ E ui uj ¼ E ðUti Uti1 ÞðUtj Utj1 Þ
¼ E Uti Utj E Uti Utj1 E Uti1 Utj þ E Uti1 Utj1
2
c2 1 ebD ebDði j 1Þ
:
¼
2b
Having modified the matrix g2V, the artificial ‘‘normal’’ distribution that
assumes i.i.d. Us that are N(0, a2) would no longer use the correct second
moment structure of the data. Thus we cannot relate a priori the
asymptotic variance of the estimator of the estimator s
^ 2 to that of the
i.i.d. normal case, as we did in Theorem 2.
9.3 Noise correlated with the price process
We have assumed so far that the U process was uncorrelated with the W
process. Microstructure noise attributable to informational effects is likely
to be correlated with the efficient price process, since it is generated by the
response of market participants to information signals (i.e., to the efficient
price process). This would be the case for instance in the bid-ask model
with adverse selection of Glosten (1987). When the U process is no longer
uncorrelated from the W process, the form of the variance matrix of the
observed log-returns Y must be altered, replacing g 2vij in Equation (26)
with
cov Yi , Yj ¼ cov sðWti Wti1 Þ þ Uti Uti1 , s Wtj Wtj1 þ Utj Utj1
¼ s2 Ddij þ cov sðWti Wti1 Þ, Utj Utj1
þ cov s Wtj Wtj1 , Uti Uti1
þ cov Uti Uti1 , Utj Utj1 ,
where dij is the Kronecker symbol.
The small sample properties of the misspecified MLE for s2 analogous
to those computed in the independent case, including its RMSE, can be
obtained from
N
2 1 X
E s
^ ¼
E Yi2
T i¼1
N
N X
i1
2
1 X
2 X
var s
^ ¼ 2
var Yi2 þ 2
cov Yi2 , Yj2 :
T i¼1
T i¼1 j¼1
Specific expressions for all these quantities depend upon the assumptions
of the particular structural model under consideration: for instance, in the
386
Sampling and Market Microstructure Noise
Glosten (1987) model (see his Proposition 6), the Us remain stationary, the
transaction noise Uti is uncorrelated with the return noise during the
previous observation period, that is, Uti1 Uti2, and the efficient return
s(Wti Wti1) is also uncorrelated with the transaction noises Uti+1 and
Uti2. With these in hand, the analysis of the RMSE and its minimum can
then proceed as above. As for the likelihood corrections for microstructure noise, the same caveat as in serially correlated U case applies: having
modified the matrix g 2V, the artificial ‘‘normal’’ distribution would no
longer use the correct second moment structure of the data and the
likelihood must be modified accordingly.
9.4 Stochastic volatility
One important departure from our basic model is the case where volatility
is stochastic. The observed log-returns are still generated by Equation (3).
Now, however, the constant volatility assumption (2) is replaced by
dXt ¼ st dWt :
ð61Þ
The object of interest in much of the literature on high frequency volatility
estimation [see, e.g., Barndorff-Nielsen and Shephard (2002), Andersen
et al. (2003)] is then the integral
Z T
s2t dt
ð62Þ
0
over a fixed time period [0,T ], or possibly several such time periods. The
estimation is based on observations 0 = t0 < t1 < < tn ¼ T, and asymptotic results are obtained when maxDti ! 0. The usual estimator for
Equation (62) is the ‘‘realized variance’’
n X
~ tiþ1 X
~ ti 2 :
X
ð63Þ
i¼1
In the context of stochastic volatility, ignoring market microstructure
noise leads to an even more dangerous situation than when s is constant
and T ! 1. We show in the companion paper Zhang, Mykland, and
Aı̈t-Sahalia (2003) that, after suitable scaling, the realized variance is a
consistent and asymptotically normal estimator — but of the quantity 2a2.
This quantity has, in general, nothing to do with the object of interest
Equation (62). Stated differently, market microstructure noise totally
swamps the variance of the price signal at the level of the realized variance.
To obtain a finite optimal sampling interval, one needs that a2 ! 0 as
n ! 1, that is, the amount of noise must disappear asymptotically. For
further developments on this topic, we refer to Zhang, Mykland, and
Aı̈t-Sahalia (2003).
387
The Review of Financial Studies / v 18 n 2 2005
10. Conclusions
We showed that the presence of market microstructure noise makes it
optimal to sample less often than would otherwise be the case in the
absence of noise, and we determined accordingly the optimal sampling
frequency in closed-form.
We then addressed the issue of what to do about it, and showed that
modeling the noise term explicitly restores the first order statistical effect
that sampling as often as possible is optimal. We also demonstrated that
this remains the case if one misspecifies the assumed distribution of the
noise term. If the econometrician assumes that the noise terms are normally distributed when in fact they are not, not only is it still optimal to
sample as often as possible, but the estimator has the same asymptotic
variance as if the noise distribution had been correctly specified. This
robustness result is, we think, a major argument in favor of incorporating
the presence of the noise when estimating continuous time models with
high frequency financial data, even if one is unsure about what is the true
distribution of the noise term. Hence, the answer to the question we pose
in our title is ‘‘as often as possible,’’ provided one accounts for the presence of the noise when designing the estimator.
Appendix A. Proof of Lemma 1
To calculate the fourth cumulant cum(Yi, Yj, Yk, Yl), recall from Equation (8) that the
observed log-returns are
Yi ¼ sðWti Wti1 Þ þ Uti Uti1 :
First, note that the t i are nonrandom, and W is independent of the Us, and has Gaussian
increments. Second, the cumulants are multilinear, so
cum Yi , Yj , Yk , Yl ¼ cum sðWti Wti1 Þ þ Uti Uti1 , s Wtj Wtj1 þ Utj Utj1 ,
sðWtk Wtk1 Þ þ Utk Utk1 , sðWtl Wtl1 Þ þ Utl Utl1 Þ
¼ s4 cum Wti Wti1 , Wtj Wtj1 , Wtk Wtk1 , Wtl Wtl1
þ s3 cum Wti Wti1 , Wtj Wtj1 , Wtk Wtk1 , Utl Utl1 ½4
2
þ s cum Wti Wti1 , Wtj Wtj1 , Utk Utk1 , Utl Utl1 ½6
þ scum Wti Wti1 , Utj Utj1 , Utk Utk1 , Utl Utl1 ½4
þ cum Uti Uti1 , Utj Utj1 , Utk Utk1 , Utl Utl1 :
Out of these terms, only the last is nonzero because W has Gaussian increments (so all
cumulants of its increments of order greater than two are zero), and is independent of the Us
(so all cumulants involving increments of both W and U are also zero). Therefore,
cum Yi , Yj , Yk , Yl ¼ cum Uti Uti1 , Utj Utj1 , Utk Utk1 , Utl Utl1 :
If i ¼ j ¼ k ¼ l, we have:
cumðUti Uti1 , Uti Uti1 , Uti Uti1 , Uti Uti1 Þ ¼ cum4 ðUti Uti1 Þ
¼ cum4 ðUti Þ þ cum4 ðUti1 Þ
¼ 2 cum4 ½U 388
Sampling and Market Microstructure Noise
with the second equality following from the independence of Uti and Uti1, and the third from
the fact that the cumulant is of even order.
If max(i, j, k, l ) ¼ min(i, j, k, l ) þ 1, two situations arise. Set m ¼ min(i, j, k, l ) and M ¼
max(i, j, k, l ). Also set s ¼ s(i, j, k, l ) ¼ #{i, j, k, l ¼ m}. If s is odd, say s ¼ 1 with i ¼ m, and
j, k, l ¼ M ¼ m þ 1, we get a term of the form
cum Utm Utm1 , Utmþ1 Utm , Utmþ1 Utm , Utmþ1 Utm ¼ cum4 ðUtm Þ:
By permutation, the same situation arises if s ¼ 3. If instead s is even, that is, s ¼ 2, then we
have terms of the form
cum Utm Utm1 , Utm Utm1 , Utmþ1 Utm , Utmþ1 Utm ¼ cum4 ðUtm Þ:
Finally, if at least one pair of indices in the quadruple (i, j, k, l ) is more than one integer apart,
then
cum Uti Uti1 , Utj Utj1 , Utk Utk1 , Utl Utl1 ¼ 0
by independence of the Us.
Appendix B. Proof of Theorem 1
Given the estimator (5) has the following expected value
N
2 1 X
N s2 D þ 2a2
2a2
¼ s2 þ
:
E s
^ ¼
E Yi2 ¼
T i¼1
T
D
The estimator’s variance is
"
#
N
N
X
2
1
1 X
2
var s
^ ¼ 2 var
Yi ¼ 2
cov Yi2 , Yj2 :
T
T i;j¼1
i¼1
Applying Lemma 1 in the special case where the first two indices and the last two respectively
are identical yields
8
> 2 cum4 ½U , if j ¼ i,
<
cum Yi , Yi , Yj , Yj ¼ cum4 ½U ,
if j ¼ i þ 1 or j ¼ i 1,
ðB:1Þ
>
:
0,
otherwise:
In the middle case, that is, whenever j ¼ i þ 1 or j ¼ i 1, the number s of indices that are
equal to the minimum index is always 2. Combining Equation (B.1) with Equation (14), we
have
N
1
N
X
2
1 X
1 N
1 X
2
2
cov Yi2 , Yi2 þ 2
cov Yi2 , Yiþ1
cov Yi2 , Yi1
þ 2
var s
^ ¼ 2
T i¼1
T i¼1
T i¼2
¼
N n
o
1 X
2 covðYi , Yi Þ2 þ 2 cum4 ½U 2
T i¼1
þ
1n
o
X
1 N
2 covðYi , Yiþ1 Þ2 þ cum4 ½U T 2 i¼1
N n
o
1 X
2 covðYi , Yi1 Þ2 þ cum4 ½U 2
T i¼2
o 2ðN 1Þ n
o
2N n
¼ 2 var½Yi 2 þ cum4 ½U þ
2 covðYi , Yi1 Þ2 þ cum4 ½U 2
T
T
þ
389
The Review of Financial Studies / v 18 n 2 2005
with var[Yi] and cov(Yi, Yi1) = cov(Yi, Yiþ1) given in Equations (9) and (10), so that
o 2ðN 1Þ 2 2N n 2
2
2a4 þ cum4 ½U var s
^ ¼ 2 s D þ 2a2 þ cum4 ½U þ
2
T
T
2 s4 D2 þ 4s2 Da2 þ 6a4 þ 2 cum4 ½U 2 2a4 þ cum4 ½U ,
¼
TD
T2
since N ¼ T/D. The expression for the RMSE follows from those for the expected value and
variance given in Equations (17) and (18):
!1=2
2
2 2a4 þ cum4 ½U 4a4 2 s4 D2 þ 4s2 Da2 þ 6a4 þ 2 cum4 ½U þ
: ðB:2Þ
RMSE s
^ ¼
TD
T2
D2
The optimal value D of the sampling interval given in Equation (19) is obtained by minimizing RMSE½^
s2 over D. The first order condition that arises from setting qRMSE½^
s2 =qD
to 0 is the cubic equation in D:
2 3a4 þ cum4 ½U 4a4 T
D
¼ 0:
ðB:3Þ
D3 4
s4
s
We now show that Equation (B.3) has a unique positive root, and that it corresponds to a
minimum of RMSE½^
s2 . We are, therefore, looking for a real positive root in D ¼ z to the
cubic equation
z3 þ pz q ¼ 0,
ðB:4Þ
where q > 0 and p < 0 since from Equation (16):
2
3a4 þ cum4 ½U ¼ 3a4 þ E U 4 3E U 2 ¼ E U 4 > 0:
Using Vièta’s change of variable from z to w given by z ¼ w p/(3w) reduces, after
multiplication by w3, the cubic to the quadratic equation
y2 qy p3
¼0
27
ðB:5Þ
in the variable y w3.
Define the discriminant
D¼
p3 q2
þ
:
3
2
The two roots of Equation (B.5) are
q
y1 ¼ þ D1=2 ,
2
y2 ¼
q
D1=2
2
are real if D 0 (and distinct if D > 0) and complex conjugates if D < 0. Then the three roots
of Equation (B.4) are
1=3
1=3
z1 ¼ y1 þ y2 ,
1 1=3
31=2 1=3
1=3
1=3
þi
y1 y2 ,
z2 ¼ y1 þ y2
2
2
1 1=3
31=2 1=3
1=3
1=3
y1 y2
i
z3 ¼ y1 þ y2
2
2
ðB:6Þ
[see, e.g., Abramowitz and Stegun (1972, Section 3.8.2)]. If D > 0, the two roots in y are both
real and positive because p < 0 and q > 0 imply
y1 > y2 > 0
and hence of the three roots given in Equation (B.6), z1 is real and positive and z2 and z3 are
complex conjugates. If D ¼ 0, then y1 ¼ y2 ¼ q/2 > 0 and the three roots are real (two of which
390
Sampling and Market Microstructure Noise
are identical) and given by
1=3
1=3
z1 ¼ y1 þ y2 ¼ 22=3 q1=3 ,
1 1=3
1
1=3
¼ z1 :
z2 ¼ z3 ¼ y1 þ y2
2
2
Of these, z1 > 0 and z2 ¼ z3 < 0. If D < 0, the three roots are distinct and real because
y1 ¼
q
þ iðDÞ1=2 reiu ,
2
y2 ¼
q
ið DÞ1=2 reiu
2
so
1=3
y1
¼ r1=3 eiu=3 ,
1=3
y2
¼ r1=3 eiu=3
and therefore
1=3
1=3
y1 þ y2
¼ 2r1=3 cosðu=3Þ,
1=3
1=3
y1 y2
¼ 2ir1=3 sinðu=3Þ
so that
z1 ¼ 2r1=3 cosðu=3Þ
z2 ¼ r1=3 cosðu=3Þ þ 31=2 r1=3 sinðu=3Þ
z3 ¼ r1=3 cosðu=3Þ 31=2 r1=3 sinðu=3Þ:
Only z1 is positive because q > 0 and (D)1/2 > 0 imply that 0 < u < p/2. Therefore
cos(u/3) > 0, so z1 > 0; sin(u/3) > 0, so z3 < 0; and
cosðu=3Þ > cosðp=6Þ ¼
31=2
¼ 31=2 sinðp=6Þ > 31=2 sinðu=3Þ,
2
so z2 < 0.
Thus Equation (B.4) has exactly one root that is positive, and it is given by z1 in
Equation (B.6). Since RMSE½^
s2 is of the form
!1=2
2
2TD3 s4 2D2 2a4 4a2 Ts2 þcum4 ½U þ2D 6a4 T þ2Tcum4 ½U þ4a4 T 2
RMSE s
^ ¼
T 2 D2
!1=2
a3 D3 þa2 D2 þa1 Dþa0
¼
T 2 D2
with a3 > 0, it tends to þ1 when D tends to þ1. Therefore, that single positive root
corresponds to a minimum of RMSE½^
s2 which is reached at
1=3
1=3
D ¼ y1 þ y2
q
1=3 q
1=3
þ D1=2
D1=2
¼
þ
:
2
2
Replacing q and p by their values in the expression above yields Equation (19). As shown
above, if the expression inside the square root in formula (19) is negative, the resulting D is
still a positive real number.
Appendix C. Proof of Proposition 1
The result follows from an application of the delta method to the known properties of the
MLE estimator of an MA(1) process [Hamilton (1995, Section 5.4)], as follows. Because we
391
The Review of Financial Studies / v 18 n 2 2005
re-use these calculations below in the proof of Theorem 2 (whose result cannot be
inferred from known MA(1) properties), we recall some of the expressions of the score vector
of the MA(1) likelihood. The partial derivatives of the log-likelihood function (25) have
the form
ih ¼ 1 q ln detðV Þ
1
qV 1
2 Y0
Y
2
qh
2g
qh
ðC:1Þ
N
1 0 1
þ
Y V Y,
2g 2 2g 4
ðC:2Þ
and
ig 2 ¼ so that the MLE for g 2 is
g
^2 ¼
1 0 1
Y V Y:
N
ðC:3Þ
At the true parameters, the expected value of the score vector is zero: E [ih] ¼ E [ig2] ¼ 0.
Hence it follows from Equation (C.1) that
2h 1 ð1 þ N Þh2N þ Nh2ð1þN Þ
qV 1
q ln detðV Þ
¼ g 2
Y ¼ g 2
E Y0
,
qh
qh
ð1 h2 Þð1 h2ð1þN Þ Þ
thus as N ! 1
qV 1
2hg2
E Y0
Y ¼
þ oð1Þ:
qh
ð1 h2 Þ
Similarly, it follows from Equation (C.2) that
E Y 0 V 1 Y ¼ Ng2 :
Turning now to Fisher’s information, we have
N
1 N
E ig2 g2 ¼ 4 þ 6 E Y 0 V 1 Y ¼ 4 ,
2g
g
2g
whence the asymptotic variance of T 1=2 ð^
g 2 g 2 Þ is 2g 4D. We also have that
h
i
1
qV 1
h
Y ¼ 2
þ oð1Þ,
E €lg2 h ¼ 4 E Y 0
2g
g ð1 h2 Þ
qh
whence the asymptotic covariance of T 1=2 ð^
g 2 g2 Þ and T 1=2 ð^
h hÞ is zero.
To evaluate E½€lhh , we compute
"
#
2 1
h
i 1 q2 ln detðV Þ
1
0q V
E €lhh ¼
þ
E
Y
Y
2
qh2
2g 2
qh2
ðC:4Þ
ðC:5Þ
ðC:6Þ
and evaluate both terms. For the first term in Equation (C.6), we have from Equation (27):
( 2 1 þ h2 þ h2þ2N 1 3h2 1 h2N
q2 ln detðV Þ
1
¼
2
2
2
qh
ð1 h2þ2N Þ
ð1 h2 Þ
)
2Nh2N 3 þ h2þ2N 4N 2 h2N
¼
392
2 1 þ h2
2
ð1 h2 Þ
þ oð1Þ:
ðC:7Þ
Sampling and Market Microstructure Noise
For the second term, we have for any non-random N N matrix Q:
E ½Y 0 QY ¼ E ½Tr½Y 0 QY ¼ E ½Tr½QYY 0 ¼ Tr½E ½QYY 0 ¼ Tr½QE ½YY 0 ¼ Tr Qg2 V ¼ g2 Tr½QV ,
where Tr denotes the matrix trace, which satisfies Tr[AB] ¼ Tr[BA]. Therefore
"
#
"
#
!
N X
N
2 1
X
q2 V 1
q2 vij
0q V
2
2
E Y
Y ¼ g Tr
V ¼g
vij
qh2
qh2
qh2
i¼1 j¼1
!
N
N
X q2 vi;iþ1
X
X
N1
q2 vii q2 vi;i1
2
1þh þ
hþ
h
¼g
qh2
qh2
qh2
i¼1
i¼1
i¼2
( 4 1 þ 2h2 þ h2þ2N 1 4h2 1 h2N
g2
¼
2
2
ð1 h2þ2N Þ
ð1 h2 Þ
)
2N 1 þ h2N 6 6h2 þ 2h2þ2N 3h4þ2N
2 2N
þ
8N
þ
h
ð1 h2 Þ
2
¼
2g2 N
þ oðN Þ:
ð1 h2 Þ
ðC:8Þ
Combining Equations (C.7) and (C.8) with Equation (C.6), it follows that
"
#
2 1
h
i 1 q2 ln detðV Þ
1
N
N
0 q VN
€
þ oðN Þ:
E lhh ¼
þ 2E Y
Y 2
N!1 ð1 h Þ
2
qh2
2g
qh2
ðC:9Þ
In light of that and Equation (C.5), the asymptotic variance of T 1=2 ð^
h hÞ is the same as in
the g2 known case, that is, (1 h2)D (which of course confirms the result of Durbin (1959) for
this parameter).
We can now retrieve the asymptotic covariance matrix for the original parameters (s2, a2)
from that of the parameters (g 2, h). This follows from the delta method applied to the change
of variable [Equations (9) and (10)]:
!
!
2 s2
D1 g2 ð1 þ hÞ2
¼
f
g
:
ðC:10Þ
,
h
¼
a2
g2 h
Hence
T 1=2
s
^2
a^2
!
s2
a2
!!
2 2 ! N 0, avar s
^ , ^a ,
T!1
where
2 0
2 2
^ ,h
^ rf g 2 , h
avar s
^ , ^a ¼ rf g2 , h avar g
0
0
1
ð1 þ hÞ2
!
ð1 þ hÞ2 2g 2 ð1 þ hÞ
B
0
2g 4 D
D
A
B
¼@
D
D
0
1 h2 D @ 2g2 ð1 þ hÞ
2
h
g
D
0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
6
2
4 s Dð4a þ s2 DÞ þ 2s4 D
s2 Dh D, s2 , a2
¼@
A:
D 2
2a þ s2 D h D, s2 , a2
2
1
h C
C
A
2
g
393
The Review of Financial Studies / v 18 n 2 2005
Appendix D. Proof of Theorem 2
We have that
Etrue ih ig2 ¼ covtrue ih , ig2
¼ covtrue N
N
1 X
qvij 1 X
, 4
Yi Yj
Yk Yl vkl
2
2g i;j¼1
qh 2g k;l¼1
!
¼
N
1 X
qvij kl
v covtrue Yi Yj , Yk Yl
6
4g i;j;k;l¼1 qh
¼
N
1 X
qvij kl v cumtrue Yi , Yj , Yk , Yl þ 2 covtrue Yi , Yj covtrue ðYk , Yl Þ ,
6
4g i;j;k;l¼1 qh
ðD:1Þ
where ‘‘true’’ denotes the true distribution of the Ys, not the incorrectly specified normal
distribution, and Cum denotes the cumulants given in Lemma 1. The last transition is because
covtrue Yi Yj , Yk Yl ¼ Etrue Yi Yj Yk Yl Etrue Yi Yj Etrue ½Yk Yl ¼ kijkl kij kkl
¼ ki;j;k;l þ ki;j kk;l ½3 ki;j kk;l
¼ ki;j;k;l þ ki;k kj;l þ ki;l kj;k
¼ cumtrue Yi , Yj , Yk , Yl þ covtrue ðYi , Yk Þcovtrue Yj , Yk
þ covtrue ðYi , Yl Þcovtrue Yj , Yk ,
since Y has mean zero [see, e.g., McCullagh (1987, Section 2.3)]. The need for permutation
goes away due to the summing over all indices (i, j, k, l ), and since V 1 ¼ [vij] is symmetric.
When looking at Equation (D.1), note that cumnormal(Yi, Yj, Yk, Yl) ¼ 0, where ‘‘normal’’
denotes a Normal distribution with the same first and second order moments as the true
distribution. That is, if the Ys were normal we would have
N
h
i
1 X
qvij kl v 2 covnormal Yi , Yj covnormal ðYk , Yl Þ :
Enormal l_h l_g2 ¼ 6
4g i;j;k;l¼1 qh
Also, since the covariance structure does not depend on Gaussianity, covtrue(Yi, Yj) ¼
covnormal(Yi, Yj). Next, we have
h
i
h i
h i
ðD:2Þ
Enormal l_h l_g2 ¼ Enormal €lhg2 ¼ Etrue €lhg2
with the last equality following from the fact that €lhg2 depends only on the second moments of
the Ys. (Note that in general Etrue ½l_h l_g2 6¼ Etrue ½€lhg2 because the likelihood may be misspecified.) Thus, it follows from Equation (D.1) that
N
h
i
h
i
1 X
qvij kl
v cumtrue Yi , Yj , Yk , Yl
Etrue l_h l_g2 ¼ Enormal l_h l_g2 6
4g i;j;k;l¼1 qh
N
h i
1 X
qvij kl
v cumtrue Yi , Yj , Yk , Yl :
¼ Etrue €lhg2 6
4g i;j;k;l¼1 qh
ðD:3Þ
It follows similarly that
Etrue
2
¼ vartrue l_h
l_h
N
h i
1 X
qvij qvkl
cumtrue Yi , Yj , Yk , Yl
¼ Etrue €lhh þ 4
4g i;j;k;l¼1 qh qh
394
ðD:4Þ
Sampling and Market Microstructure Noise
and
Etrue
2
l_g2
¼ vartrue l_g2
N
h
i
1 X
vij vkl cumtrue Yi , Yj , Yk , Yl :
¼ Etrue €lg2 g2 þ 8
4g i;j;k;l¼1
ðD:5Þ
We now need to evaluate the sums that appear on the right-hand sides of Equations (D.3)–
(D.5). Consider two generic symmetric N N matrices [ni, j] and [vi, j]. We are interested in
expressions of the form
X
ð1Þs ni;j vk;l ¼
X
N
1
X
ð1Þs ni;j vk;l
h¼1 i;j;k;l:m¼h;M¼hþ1
i;j;k;l:M¼mþ1
¼
N
1X
3
X
X
ð1Þr ni;j vk;l
h¼1 r¼1 i;j;k;l:m¼h;M¼hþ1;s¼r
¼
N
1n
X
2nh;hþ1 vhþ1;hþ1 2nhþ1;hþ1 vh;hþ1
h¼1
þ nh;h vhþ1;hþ1 þ nhþ1;hþ1 vh;h þ 4nh;hþ1 vh;hþ1
o
2nhþ1;h vh;h 2nh;h vhþ1;h :
ðD:6Þ
It follows that if we set
Yðn, vÞ ¼
N
X
ni;j vk;l cumtrue Yi , Yj , Yk , Yl ,
ðD:7Þ
i;j;k;l¼1
then Y(n, v) ¼ cum4[U] C (n, v) where
N
N
1 n
X
X
2nh;hþ1 vhþ1;hþ1 2nhþ1;hþ1 vh;hþ1
cðn, vÞ ¼ 2 nh;h vh;h þ
h¼1
h¼1
þ nh;h vhþ1;hþ1 þ nhþ1;hþ1 vh;h þ 4nh;hþ1 vh;hþ1
o
2nhþ1;h vh;h 2nh;h vhþ1;h :
ðD:8Þ
If the two matrices [ni,j] and [vi,j ] satisfy the following reversibility property:
nNþ1i,Nþ1j ¼ ni,j and vNþ1i,Nþ1j ¼ vi,j (so long as one is within the index set), then
Equation (D.8) simplifies to:
N
N
1 n
X
X
cðn, vÞ ¼ 2 nh;h vh;h þ
4nh;hþ1 vhþ1;hþ1 4nhþ1;hþ1 vh;hþ1
h¼1
h¼1
o
þ 2nh;h vhþ1;hþ1 þ 4nh;hþ1 vh;hþ1 :
This is the case for V 1 and its derivative qV 1/qh, as can be seen from the expression for vi,j
given in Equation (28), and consequently for qvi,j/qh.
Therefore, if we wish to compute the sums in Equations (D.3)–(D.5) we need to find the
three quantities c(qv/qh,v),c(qv/qh,qv/qh), and c(v,v), respectively. All are of order O(N), and
only the first term is needed. Replacing the terms vi,j and qvi,j/qh by their expressions from
395
The Review of Financial Studies / v 18 n 2 2005
Equation (28), we obtain:
cðv, vÞ ¼
¼
2
3
2
ð1 h2ð1þN Þ Þ
n
ð1 þ hÞ 1 h2N 1 þ 2h2 þ 2h2ð1þN Þ þ h2ð2þN Þ
ð1 þ h2 Þð1 hÞ
o
þ N ð1 hÞ 1 þ h2 2 þ h2N þ h2ð1þN Þ þ 6h1þ2N þ 2h2þ4N
4N
þ oðN Þ,
ð1 hÞ2
ðD:9Þ
2 Oð1Þ þ 2N ð1 hÞ 1 þ h2 h 1 þ h2 þ O h2N þ N 2 O h2N
qv
,v ¼
c
3
2
qh
hð1 hÞ4 ð1 þ h2 Þ ð1 h2ð1þN Þ Þ
¼
4N
ð1 hÞ3
þ oðN Þ,
ðD:10Þ
4 Oð1Þ þ 3N 1 h4 h2 1 þ h2 2 þ Oh2N þ N 2 Oh2N þ N 3 Oh2N qv qv
,
¼
c
4
3
qh qh
3h2 ð1 þ hÞð1 þ h2 Þ ð1 hÞ5 ð1 h2ð1þN Þ Þ
¼
4N
ð1 hÞ4
þ oðN Þ:
ðD:11Þ
^ Þ obtained by maximizing the (incorrectlyThe asymptotic variance of the estimator ð^
g2 , h
specified) log-likelihood function (25) that assumes Gaussianity of the Us is given by
2 1
^ ,h
^ ¼ D D0 S1 D
avartrue g
where, from Equations (C.4), (C.5), and (C.9) we have
h i
hi
hi 1
1
1
D ¼ D0 ¼ Etrue €l ¼ Enormal €l ¼ Enormal l_l_0
N
N
N
1
0
1
h
1
B 2g 4 Ng 2 ð1 h2 Þ þ o N C
B
C
¼@
A
1
þ
o
ð
1
Þ
ð1 h2 Þ
ðD:12Þ
and, in light of Equations (D.3)–(D.5),
S¼
where
h i
hi
1
1
Etrue l_l_0 ¼ Etrue €l þ cum4 ½U C ¼ D þ cum4 ½U C,
N
N
1
1
1
qv
,
v
c
ð
v,
v
Þ
c
8
1 B
g 6
qh C
Bg
C
C¼
1
qv qv A
4N @
,
c
g4
qh qh
0
1
1
1
þ
o
ð
1
Þ
þ oð1Þ C
B 8
2
3
g 6 ð1 hÞ
B g ð1 hÞ
C
¼B
C
1
@
A
þ oð1Þ
4
4
g ð1 hÞ
ðD:13Þ
0
396
ðD:14Þ
Sampling and Market Microstructure Noise
from the expressions just computed. It follows that
1
2 ^ ,h
^ ¼ D DðD þ cum4 ½U CÞ1 D
avartrue g
1 1
¼ D D Id þ cum4 ½U D1 C
¼ D Id þ cum4 ½U D1 C D1
¼ D Id þ cum4 ½U D1 C D1
2 ¼ avarnormal g
^ ,h
^ þ Dcum4 ½U D1 CD1 ,
where Id denotes the identity matrix and
2 ^ ,h
^ ¼
avarnormal g
2g 4 D
0
!
0
0
,
1 h2 D
2ð1 þ hÞ
4
B ð1 hÞ
B
D1 CD1 ¼ B
@
2
1
g 2 ð1 hÞ C
C
C
ð1 þ hÞ2 A
2
g 4 ð1 hÞ2
so that
2 2g 4
^ ,h
avartrue g
^ ¼D
0
!
0
2
1h
0
2ð1 þ hÞ
4
B ð1 hÞ
B
þ Dcum4 ½U B
@
2
1
g 2 ð1 hÞ C
C
C:
ð1 þ hÞ2 A
2
g 4 ð1 hÞ2
By applying the delta method to change the parametrization, we now recover the asymptotic
variance of the estimates of the original parameters:
2 0
2 2
^ , ^a ¼ rf g 2 , h avartrue g
^ ,h
^ rf g2 , h
avartrue s
0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
4 s6 Dð4a2 þ s2 DÞ þ 2s4 D
s2 Dh D, s2 , a2
B
C
C:
¼B
@
A
D 2
2
2 2
2a þ s D h D, s , a þ Dcum4 ½U 2
Appendix E. Derivations for Section 7
To see Equation (39), let ‘‘orig’’ (E.7) denote parametrization in (and differentiation with
respect to) the original parameters s2 and a2, while ‘‘transf’’ denotes parametrization and
differentiation in g2 and h, and finv denotes the inverse of the change of variable function
defined in (C.10), namely
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
1
!
2a2 þ s2 D þ s2 Dð4a2 þ s2 DÞ
B
C
2
B 2
g
C
C
ðE:1Þ
¼ finv s2 , a2 ¼ B
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
@ 1
A
h
2
2
2 Dð4a2 þ s2 DÞ
2a
s
D
þ
s
2
2a
and rfinv its Jacobian matrix. Then, from l_orig ¼ rfinv ðs2 ; a2 Þ0 l_transf , we have
h
i
0
€lorig ¼ rfinv s2 , a2 €ltransf rfinv s2 , a2 þ H l_transf ,
where H½l_transf is a 2 2 matrix whose terms are linear in l_transf and the second partial
derivatives of finv. Now Etrue ½l_orig ¼ Etrue ½l_transf ¼ 0, and so Etrue ½H½l_transf ¼ 0 from which it
397
The Review of Financial Studies / v 18 n 2 2005
follows that
h
i
Dorig ¼ N 1 Etrue €lorig
0
¼ rfinv s2 , a2 Dtransf rfinv s2 , a2
0 1=2 2
1
D
2a þ s2 D
D1=2
B 3 2
C
3=2
3=2
B 2s ð4a þ s2 DÞ
C
sð4a2 þ s2 DÞ
C þ oð1Þ
!
¼B
B
C
1=2
2
2
D s 6a þ s D A
1
@
1
4
3=2
2
2
2a
ð4a þ s DÞ
with Dtransf ¼ N 1 Etrue ½€ltransf given in Equation
0
rfinv ðs2 , a2 Þ and so
rfinv ðs2 , a2 Þ0 l_transf l_transf
(D.12).
Similarly,
0
Sorig ¼ rfinv s2 , a2 Stransf rfinv s2 , a2
0
¼ rfinv s2 , a2 ðDtransf þ cum4 ½U CÞ rfinv s2 , a2
2 2 0
2 2
¼ Dorig þ cum4 ½U rfinv s , a C rfinv s , a
ðE:2Þ
0
l_orig l_orig
¼
ðE:3Þ
with the second equality following from the expression for Stransf given in Equation (D.13).
To complete the calculation, note from Equation (D.14) that
0
þ oð1Þ,
C ¼ gtransf gtransf
where
gtransf ¼
g 4 ð1 hÞ1
g 2 ð1 hÞ2
!
:
Thus
0
0
þ oð1Þ,
rfinv s2 , a2 C rfinv s2 , a2 ¼ gorig gorig
ðE:4Þ
where
0
B
B
0
g ¼ gorig ¼ rfinv s2 , a2 gtransf ¼ B
B
@ 1
2a4
D1=2
1
C
3=2
C
sð4a2 þ s2 DÞ
2
! C
C
1=2
2
D s 6a þ s D A
1
3=2
ð4a2 þ s2 DÞ
ðE:5Þ
which is the result (40). Inserting Equation (E.4) into Equation (E.3) yields the result (39).
For the profile likelihood l, let ^a2s2 denote the maximizer of l(s2, a2) for given s2. Thus by
definition lðs2 Þ ¼ lðs2 , ^a2s2 Þ. From now on, all differentiation takes place with respect to
the original parameters, and we will omit the subscript ‘‘orig’’ in what follows. Since
0 ¼ l_a2 ðs2 , ^a2s2 Þ, it follows that
q _ 2 2 l 2 s , ^as2
qs2 a
q^a2s2
¼ €ls2 a2 s2 , ^a2s2 þ €la2 a2 s2 , ^a2s2
,
qs2
0¼
so that
€ls2 a2 s2 , a^2 2
q^a2s2
s
¼
:
€la2 a2 s2 , ^a2 2
qs2
s
398
ðE:6Þ
Sampling and Market Microstructure Noise
The profile score then follows
q^a2s2
l_ s2 ¼ l_s2 s2 , ^a2s2 þ l_a2 s2 , ^a2s2
qs2
ðE:7Þ
so that at the true value of (s2, a2),
l_ s
2
¼ l_s2
h
i
Etrue €ls2 a2
h
i l_a2 s2 , a2 þ Op ð1Þ,
s ,a Etrue €la2 a2
2
2
ðE:8Þ
since ^
a2 ¼ a2 þ Op ðN 1=2 Þ and
h
i
D€ls2 a2 N 1€ls2 a2 s2 , ^a2s2 N 1 Etrue €ls2 a2 ¼ Op N 1=2
h
i
D€la2 a2 N 1€la2 a2 s2 , ^a2s2 N 1 Etrue €la2 a2 ¼ Op N 1=2
as sums of random variables with expected value zero, so that
q^a2 2 N 1€ls2 a2 s2 , ^a2s2
s2 ¼
qs
N 1€la2 a2 s2 , ^a2s2
h
i
N 1 Etrue €ls2 a2 þ D€ls2 a2
h
i
¼
N 1 Etrue €la2 a2 þ D€la2 a2
h
i
Etrue €ls2 a2
h
i þ D€ls2 a2 D€la2 a2 þ op N 1=2
¼
€
Etrue la2 a2
h
i
Etrue €ls2 a2
h
i þ Op N 1=2 ,
¼
Etrue €la2 a2
while
l_a2 s2 , a2 ¼ Op N 1=2
also as a sum of random variables with expected value zero.
Therefore
h
i
h h 2 i Etrue €ls2 a2
i
2
2
h
i Etrue l_a2 s2 , a2 þ Oð1Þ
¼ Etrue l_s2 s , a Etrue l_ s
Etrue €la2 a2
¼ Oð1Þ,
since Etrue ½l_s2 ðs2 , a2 Þ ¼ Etrue ½l_a2 ðs2 , a2 Þ ¼ 0. In particular, Etrue ½l_ ðs2 Þ ¼ oðNÞ as claimed.
Further differentiating Equation (E.7), one obtains
2
q^a2s2
€ s2 ¼ €ls2 s2 s2 , ^a2 2 þ €la2 a2 s2 , ^a2 2
l
s
s
qs2
q^a2s2
2 2
_a2 s2 , ^a2 2 q ^as2
þ 2€ls2 a2 s2 , ^a2s2
þ
l
s
qs2
q2 s2
2
2 2 €ls2 a2 s2 , ^a2s2
2 2 q2 ^a2s2
¼ €ls2 s2 s , ^as2 þ l_a2 s , ^as2 2
€la2 a2 s2 , ^a2 2
q s2
s
399
The Review of Financial Studies / v 18 n 2 2005
from Equation (E.6). Evaluated at s2 ¼ s
^ 2 , one gets ^a2s2 ¼ ^a2 and l_a2 ð^
s2 , ^a2 Þ ¼ 0, and so
2 2 2
2
2 2 €ls2 a2 s
^ , ^a
€ s
l
^ ¼ €ls2 s2 s
^ , ^a €la2 a2 ðs
^ 2 , ^a2 Þ
1
i
¼h
,
1
€l ðs
^ 2 , ^a2 Þ
ðE:9Þ
s2 s 2
where ½€l ð^
s2 , ^a2 Þ1 s2 s2 is the upper left element of the matrix €l ð^
s2 , a^2 Þ1 . Thus Equation (42)
is valid.
Alternatively, we can see that the profile likelihood l satisfies the Bartlett identity to first
order, that is, Equation (43). Note that by Equation (E.8),
20
h
i
12 3
€ls2 a2
h 2 i
E
true
6
h
i l_a2 s2 , a2 þ Op ð1ÞA 7
¼ N 1 Etrue 4@l_s2 s2 , a2 N 1 Etrue l_ s2
5
Etrue €la2 a2
20
h
i
12 3
€ls2 a2
E
true
6
h
i l_a2 s2 , a2 A 7
¼ N 1 Etrue 4@l_s2 s2 , a2 5 þ oð1Þ
Etrue €la2 a2
2
h
i
0
12
€
6_ 2 2 2 @Etrue hls2 a2 i _ 2 2 A
1
la2 s , a
¼ N Etrue 4ls2 s , a þ
Etrue €la2 a2
h
i
3
Etrue €ls2 a2
2 2 2 2
_
_
h
i la2 s , a ls2 s , a 5 þ oð1Þ
2
Etrue €la2 a2
so that
h 2 i
Ds2 a2 2
D 2 2
N 1 Etrue l_ s2
¼ Ss 2 s 2 þ
Sa2 a2 2 s a Sa2 s2 þ op ð1Þ
Da2 a2
Da2 a2
!
Ds2 a2 2
Ds2 a2
Da2 a2 2
D2 2
¼ Ds2 s2 þ
Da2 a2
Da2 a2 a s
!
Ds2 a2 2 2
D 2 2
ga2 2 s a gs2 ga2 þ op ð1Þ
þ cum4 ½U gs2 2 þ
Da2 a2
Da2 a2
by invoking Equation (39).
Continuing the calculation,
2
h 2 i D2 2 2
D 2 2
¼ Ds2 s2 s a þ cum4 ½U gs2 s a ga2 þ oð1Þ
N 1 Etrue l_ s2
Da2 a2
Da2 a2
¼ 1= D1 s2 s2 þ oð1Þ,
ðE:10Þ
since from the expressions for Dorig and gorig in Equations (E.2) and (E.5) we have
gs2 Ds2 a2
g 2 ¼ 0:
Da2 a2 a
Then by Equation (E.9) and the law of large numbers, we have
2 € s
¼ 1= D1 s2 s2 þ oð1Þ,
N 1 Etrue l
and Equation (43) follows from combining Equation (E.10) with Equation (E.12).
400
ðE:11Þ
ðE:12Þ
Sampling and Market Microstructure Noise
Appendix F. Proof of Lemma 2
1 Id implies that
q1
q 1
¼ 1
qb1
qb1
ðF:1Þ
and, since is linear in the parameters s2 and a2 (see Equation (45)) we have
q2 ¼0
qb2 qb1
ðF:2Þ
so that
q2 1
q q1
¼
qb2 qb1 qb2 qb1
q 1 q 1
q 1 q 1
q2 þ 1
1
1
qb2
qb1
qb1
qb2
qb1 qb2
q 1 q 1
q 1 q 1
¼ 1
þ 1
:
qb2
qb1
qb1
qb2
¼ 1
ðF:3Þ
In the rest of this lemma, let expectations be conditional on the Ds. We use the notation
E [ jD] as a shortcut for E [ jDN, . . . , D1]. At the true value of the parameter vector, we have,
h
i
0 ¼ E l_b1 jD
1 q ln det 1
q1
E Y0
Y jD :
ðF:4Þ
¼
2 qb1
2
qb1
with the second equality following from Equation (46). Then, for any nonrandom Q, we have
E ½Y 0 QY ¼ Tr½QE ½YY 0 ¼ Tr½Q:
ðF:5Þ
This can be applied to Q that depends on the Ds, even when they are random, because the
expected value is conditional on the Ds. Therefore, it follows from Equation (F.4) that
1 1
q ln det q
q
0 q
¼ E Y
Y jD ¼ Tr
¼ Tr 1
,
ðF:6Þ
qb1
qb1
qb1
qb1
with the last equality following from Equation (F.1) and so
q2 ln det
q
q
¼
Tr 1
qb2 qb1
qb2
qb1
q
1 q
¼ Tr
qb2
qb1
"
#
q 1 q
q2 þ 1
¼ Tr 1
qb2
qb1
qb2 qb1
q 1 q
¼ Tr 1
,
qb2
qb1
ðF:7Þ
again because of Equation (F.2).
In light of Equation (46), the expected information (conditional on the Ds) is given by
"
#
2 1
h
i 1 q2 ln det 1
0q €
þ E Y
Y jD :
E lb2 b1 jD ¼
2 qb2 b1
2
qb2 b1
401
The Review of Financial Studies / v 18 n 2 2005
Then,
"
#
"
#
q2 1
q2 1
E Y
Y jD ¼ Tr
qb2 b1
qb2 b1
q 1 q
q 1 q
þ 1
¼ Tr 1
qb2
qb1
qb1
qb2
q
q
¼ 2Tr 1
1
qb2
qb1
0
with the first equality following from Equation (F.5) applied to Q ¼ q21/qb2b1, the second
from Equation (F.3), and the third from the fact that Tr[AB] ¼ Tr[BA]. It follows that
h
i
1
q 1 q
q 1 q
þ Tr 1
E €lb2 b1 jD ¼ Tr 1
2
qb2
qb1
qb2
qb1
1
1 q 1 q
¼ Tr 2
qb2
qb1
¼
1 q2 ln det
:
2 qb2 b1
Appendix G. Proof of Theorem 3
In light of Equations (45) and (52),
¼ 0 þ es2 X
ðG:1Þ
from which it follows that
1
1 ¼ 0 Id þ es2 1
0 X
1 1
¼ Id þ es2 1
0
0 X
1 2 1
3
2 1
1
2 4
¼ 1
0 es 0 X0 þ e s 0 X 0 þ O e ,
ðG:2Þ
since
ðId þ eAÞ1 ¼ Id eA þ e2 A2 þ O e3 :
Also,
q q0
qs2
¼
þe
X:
qb1 qb1
qb1
Therefore, recalling Equation (F.6), we have
q ln det q
¼ Tr 1
qb1
qb1
1 2 1
3 q0
qs2
2 1
1
2 4
þe
X
¼ Tr 1
0 es 0 X0 þ e s 0 X 0 þ O e
qb1
qb1
2
q
q
qs
0
0
1
þ
1 X
þ eTr s2 1
¼ Tr 1
0
0 X0
qb1
qb1
qb1 0
2
2 1 q0
2 qs
1
1
þ e2 Tr s4 1
X
s
X
X
0
0
0
qb1
qb1 0
3
þ Op e :
ðG:3Þ
402
Sampling and Market Microstructure Noise
We now consider the behavior as N ! 1 of the terms up to order e2. The remainder term is
handled similarly. Two things can be determined from the above expansion. Since the jis are
i.i.d. with mean 0, E [X] ¼ 0, and so, taking unconditional expectations with respect to the law
of the Dis, we obtain that the coefficient of order e is
qs2 1
1 q0
E Tr s2 1
þ
0 X
0 X0
qb1
qb1
qs2 1
2 1
1 q0
þ
0 X
¼ Tr E s 0 X0
qb1
qb1
q
qs2 1
0
1
þ
0 E ½X
¼ Tr s2 1
0 E ½X0
qb1
qb1
¼ 0:
Similarly, the coefficient of order e2 is
2 1 q0
qs2 1 2
E Tr s4 1
s2
0 X
0 X 0
qb1
qb1
h
i
q
qs2 h 1 2 i
2
0
1
s2
E 0 X
¼ Tr s4 E 1
0 X
0
qb1
qb1
h
2 i 2 1 q0 qs2
2
1
s 0
Id
¼ Tr s E 0 X
qb1
qb1
q
qs2
0
1
¼ Tr s2 1
s2 1
Id :
0 E X0 X
0
qb1
qb1
The matrix E½X1
0 X has the following terms
N X
N
X
1 1 2
X0 X i;j ¼
Xik 1
0 kl Xlj ¼ D0 j i j j 0 ij
k¼1 l¼1
and since E [jijj] ¼ dij var[j] (where dij denotes the Kronecker symbol), it follows that
1 2
E X1
ðG:4Þ
0 X ¼ D0 var½j diag 0 ,
1
where diag½1
0 is the diagonal matrix formed with the diagonal elements of 0 . From this,
we obtain that
1 2 1 q0 qs2
q ln det q0
2
2 1
Tr
s
E
X
X
s
Id
þ O e3
¼ Tr 1
þ
e
E
0
0
0
0
qb1
qb1
qb1
qb1
q0
¼ Tr 1
0
qb1
1 2 1 q0 qs2
Id þ O e3 : ðG:5Þ
s 0
þ e2 D20 var½jTr s2 1
0 diag 0
qb1
qb1
To calculate E½€lb2 b1 , in light of Equation (51), we need to differentiate E [q ln det /qb1]
with respect to b2. Indeed
"
#
h
i
h h
ii
1 q2 ln det 1 q
q ln det E €lb2 b1 ¼ E E €lb2 b1 jD ¼ E
,
E
¼
2
qb2 qb1
2 qb2
qb1
where we can interchange the unconditional expectation and the differentiation with respect
to b2 because the unconditional expectation is taken with respect to the law of the Dis, which
is independent of the b parameters (i.e., s2 and a2). Therefore, differentiating Equation (G.5)
403
The Review of Financial Studies / v 18 n 2 2005
with respect to b2 will produce the result we need. (The reader may wonder why we take the
expected value before differentiating, rather than the other way around. As just discussed, the
results are identical. However, it turns out that taking expectations first reduces the computational burden quite substantially.)
Combining with Equation (G.5), we therefore have
h
i
1 q
q ln det E
E €lb2 b1 ¼ 2 qb2
qb1
1 q
1 q0
¼
Tr 0
2 qb2
qb1
1 2 1 q0 qs2
1 2 2
q
Tr s2 1
Id
þ O e3
s 0
e D0 var½j
0 diag 0
2
qb2
qb1
qb1
ðG:6Þ
fð0Þ þ e2 fð2Þ þ O e3 :
It is useful now to introduce the same transformed parameters (g 2, h) as in previous sections
and write 0 ¼ g 2V with the parameters and V defined as in Equations (9), (10), and (26),
except that D is replaced by D0 in these expressions. To compute f(0), we start with
2 q0
2 1 q g V
Tr 1
V
¼
Tr
g
0
qb1
qb1
qg 2
1 qV
þ Tr g2 V 1 V
¼ Tr V
qb1
qb1
2
qV
qh
qg
þ Ng2
ðG:7Þ
¼ Tr V 1
qh qb1
qb1
with qg 2/qb1 and qh/qb1 to be computed from Equations (11) and (12). If Id denotes the
identity matrix and J the matrix with 1 on the infra and supra-diagonal lines and 0
everywhere else, we have V ¼ h2Id þ hJ, so that qV/qh ¼ 2hIdþJ. Therefore
qV
¼ 2hTr V 1 þ Tr V 1 J
Tr V 1
qh
N
N
1
X
X
vi;i1 þ vi;iþ1 þ v1;2 þ vN;N1
¼ 2h vi;i þ
i¼1
i¼2
2h 1 h2N N 1 h2 þ 1
¼
ð1 h2 Þð1 h2ð1þN Þ Þ
2h
þ oð1Þ:
¼
ð1 h2 Þ
ðG:8Þ
Therefore, the first term in Equation (G.7) is O(1) while the second term is O(N) and hence
q0
qg 2
Tr 1
þ Oð1Þ:
¼ Ng 2
0
qb1
qb1
This holds also for the partial derivative of Equation (G.7) with respect to b2. Indeed, given
the form of Equation (G.8), we have that
q
qV
q
2h
¼
þ oð1Þ ¼ Oð1Þ,
Tr V 1
2
qb2
qh
qb2 ð1 h Þ
since the remainder term in Equation (G.8) is of the form p(N)hq(N), where p and q are
polynomials in N or order greater than or equal to 0 and 1, respectively, whose differentiation
404
Sampling and Market Microstructure Noise
with respect to h will produce terms that are of order o(N). Thus it follows that
q
q0
q
qg 2
Tr 1
g 2
¼N
þ oðN Þ
0
qb2
qb
qb1
qb1
( 2
)
qg 2 qg 2
q2 g 2
þ g 2
¼N
þ oðN Þ:
qb2 qb1
qb2 qb1
ðG:9Þ
Writing the result in matrix form, where the (1, 1) element corresponds to (b1, b2) = (s2, s2),
the (1, 2) and (2, 1) elements to (b1, b2) ¼ (s2, a2) and the (2, 2) element to (b1, b2) ¼ (a2, a2),
and computing the partial derivatives in Equation (G.9), we have
1 q
q0
fð0Þ ¼ Tr 1
0
2 qb
qb1
0 21=2 1
1=2
2
D0 2a þ s2 D0
D0
B 3 2
C
3=2
B 2s ð4a þ s2 D0 Þ3=2
C
sð4a2 þ s2 D0 Þ
C þ oðN Þ:
!
¼ NB
ðG:10Þ
1=2
B
2
2
D0 s 6a þ s D0 C
1
@
A
1
3=2
2a4
ð4a2 þ s2 D0 Þ
As for the coefficient of order e2, that is f(2) in Equation (G.6), define
1 2 1 q0 qs2
Id
s 0
a Tr s2 1
0 diag 0
qb1
qb1
ðG:11Þ
so that
1
qa
fð2Þ ¼ D20 var½j
:
2
qb2
We have
1 1 q0
qs2 1
diag
Tr 0 diag 1
s2
a ¼ s4 Tr 1
0
0
0
0
qb1 qb
1
q g2 V
qs2 4 1
¼ s4 g 6 Tr V 1 diag V 1 V 1
g Tr V diag V 1
s2
qb1
qb1
qV qh
qg 2 1
¼ s4 g 4 Tr V 1 diag V 1 V 1
Tr V diag V 1
þ s4 g 6
qh qb1
qb1
qs2 4 1
s2
g Tr V diag V 1
qb1
qh
qV
qg2
qs2 4
þ s4 g 6
Tr V 1 diag V 1 V 1
s2
g
Tr V 1 diag V 1 :
¼ s4 g 4
qb1
qh
qb1
qb1
Next, we compute separately
qV
qV 1
¼ Tr diag V 1 V 1
V
Tr V 1 diag V 1 V 1
qh
qh
1 qV 1
¼ Tr diag V
qh
N
i;i
X
qv
¼ vi;i
qh
i¼1
Oð1Þ 2Nh 1 þ h2 h4 h6 þ O h2N þ N 2 O h2N
¼
3
2
4
ð1 þ h2 Þ ð1 h2 Þ ð1 h2ð1þN Þ Þ
2Nh
¼
þ oðN Þ
3
ð1 h2 Þ
405
The Review of Financial Studies / v 18 n 2 2005
and
N X
2
Tr V 1 diag V 1 ¼
vi;i
i¼1
¼
Oð1Þ þ N 1 h4 þ O h2N
3
2
ð1 þ h2 Þð1 h2 Þ ð1 h2ð1þN Þ Þ
N
¼
þ oðN Þ:
2
ð1 h2 Þ
Therefore
a ¼ s4 g4
qh
qb1
!
2Nh
3
ð1 h2 Þ
!
qg2
qs2 4
N
s2
g
þ s4 g6
þ oðN Þ,
2
qb1
qb1
ð1 h2 Þ
which can be differentiated with respect to b2 to produce qa/qb2. As above, differentiation of
the remainder term o(N) still produces a o(N) term because of the structure of the terms there
(they are again of the form p(N)hq(N)).
Note that an alternative expression for a can be obtained as follows. Going back to the
definition (G.11),
1 1 q0
qs2 1
diag
Tr 0 diag 1
s2
ðG:12Þ
a ¼ s4 Tr 1
0
0
0
0
qb1
qb1
the first trace becomes
1 1 q0
1 q0 1
Tr 1
0
¼ Tr diag 1
0 diag 0 0
0 0
qb1
qb1
1 q1
0
¼ Tr diag 0
qb1
1 N X
q0
¼
1
0
ii
qb1 ii
i¼1
¼
N 2
1 q X
1
2 qb1 i¼1 0 ii
¼
1 1 q
,
Tr 1
0 diag 0
2 qb1
so that we have
1 1 q
qs2 1
Tr 1
Tr 0 diag 1
s2
0 diag 0
0
2 qb1
qb1
1 1 qs4
1 1 q
¼ s4
Tr 1
Tr 1
0 diag 0
0 diag 0
2 qb1
2 qb1
1 q 4 1
¼
s Tr 0 diag 1
0
2 qb1
1 q 4 4 1
s g Tr V diag V 1
¼
2 qb1
!!
1 q
N
4 4
¼
s g
þ oðN Þ
2
2 qb1
ð1 h2 Þ
!
N q
s4 g 4
¼
þ oðN Þ,
2 qb1 ð1 h2 Þ2
a ¼ s4
406
Sampling and Market Microstructure Noise
where the calculation of Tr[V 1diag[V 1]] is as before, and where the o(N) term is a sum of
terms of the form p(N)hq(N) as discussed above. From this one can interchange differentiation
and the o(N) term, yielding the final equality above.
Therefore
!!
qa
1 q2
N
4 4
¼
s g
þ oðN Þ
2
qb2
2 qb1 qb2
ð1 h2 Þ
!
N q2
s4 g 4
¼
þ oðN Þ:
ðG:13Þ
2 qb1 qb2 ð1 h2 Þ2
Writing the result in matrix form and calculating the partial derivatives, we obtain
0
2
1
8a 2s2 D0
2
2a C
1
qa
ND20 var½j B
2D0
B
C
¼
fð2Þ ¼ D20 var½j
B
C þ oðN Þ:
3
2
2
2
A
2
qb2 ð4a þ s D0 Þ @
8s
D0
Putting it all together, we have obtained
1 h € i 1 ð0Þ
E lb2 b1 ¼
f þ e2 fð2Þ þ O e3
N
N
F ð0Þ þ e2 F ð2Þ þ O e3 þ oð1Þ,
where
0
1=2 D0
2a2 þ s2 D0
F ð2Þ 1
1=2
C
3=2
C
sð4a2 þ s2 D0 Þ
C
!
C,
1=2
2
2
D0 s 6a þ s D0 C
A
1
3=2
ð4a2 þ s2 D0 Þ
1
2a4
0
D20 var½j
3
ð4a2 þ s2 D0 Þ
ðG:15Þ
D0
B
B 2s3 ð4a2 þ s2 D Þ3=2
0
B
F ð0Þ ¼ B
B
@
B 2a
B
B
@
2
8a2 2s2 D0
2D0
8s2
D0
ðG:14Þ
ðG:16Þ
1
C
C
C:
A
ðG:17Þ
The asymptotic variance of the maximum-likelihood estimators avarð^
s2 , ^a2 Þ is, therefore,
given by
2 2
1
avar s
^ , ^a ¼ E ½D F ð0Þ þ e2 F ð2Þ þ O e3
h
i1
1
¼ D0 F ð0Þ Id þ e2 F ð0Þ F ð2Þ þ O e3
h
i1
1 h ð0Þ i1
¼ D0 Id þ e2 F ð0Þ F ð2Þ þ O e3
F
h
i1
h ð0Þ i1
¼ D0 Id e2 F ð0Þ F ð2Þ þ O e3
F
h
i1
h
i1
h
i1
¼ D0 F ð0Þ
e2 D0 F ð0Þ F ð2Þ F ð0Þ
þ O e3
Að0Þ þ e2 Að2Þ þ O e3 ,
where the final results for A(0) ¼ D0[F (0)]1 and A(2) ¼ D0[F (0)]1F (2)[F (0)]1, obtained by
replacing F (0) and F (2) by their expressions in Equation (G.15), are given in the statement of
the theorem.
407
The Review of Financial Studies / v 18 n 2 2005
Appendix H. Proof of Theorem 4
It follows as in Equations (D.3)–(D.5) that
h
i
Etrue l_b1 l_b2 jD ¼ covtrue l_b1 , l_b2 jD
1 1 !
N
N
1X
q
1X
q
Yi Yj
,
Yk Yl
¼ covtrue 2 i;j¼1
2 k;l¼1
qb1 ij
qb2 kl
N
1 X q1
q1
covtrue Yi Yj , Yk Yl jD
¼
4 i;j;k;l¼1 qb1 ij qb2 kl
1 N h
i 1 X
q1
q
¼ Etrue €lb1 b2 jD þ
cumtrue Yi , Yj , Yk , Yl jD
4 i;j;k;l¼1 qb1 ij qb2 kl
1
h
i 1
q
q1
¼ Etrue €lb1 b2 jD þ cum4 ½U c
,
,
ðH:1Þ
4
qb1 qb2
since cumtrue(Yi, Yj, Yk, YljD) ¼ 2, 1, or 0, cumtrue(U), as in Equation (15), and with c
defined in Equation (D.8). Taking now unconditional expectations, we have
h
i
Etrue l_b1 l_b2 ¼ covtrue l_b1 , l_b2
h
i
h
i
h
i
¼ E covtrue l_b1 , l_b2 jD þ covtrue Etrue l_b1 jD , Etrue l_b2 jD
h
i
¼ E covtrue l_b1 , l_b2 jD
1
h
i 1
q
q1
,
:
ðH:2Þ
¼ Etrue €lb1 b2 þ cum4 ½U E c
4
qb1 qb2
with the first and third equalities following from the fact that Etrue ½l_bi jD ¼ 0.
Since
h
i
h
i
Etrue €lb1 b2 jD ¼ Enormal €lb1 b2 jD
and consequently
h
i
h
i
Etrue €lb1 b2 ¼ Enormal €lb1 b2
have been found in the previous subsection (see Equation (G.15)), what we need to do to
obtain Etrue ½l_b1 l_b2 is to calculate
1
q
q1
E c
,
:
qb1 qb2
With 1 given by Equation (G.2), we have for i ¼ 1, 2
q1 q1
q 2 1 1 q 4 1 2 1 ¼ 0 e
s 0 X0 þ e2
s 0 X 0 þ O e3
qbi
qbi
qbi
qbi
and, therefore, by bilinearity of c we have
1
1
1
q
q1
q0 q1
q0
q 2 1 1 0
,
,
,
s 0 X0
¼c
eE c
½2
E c
qb1 qb2
qb1 qb2
qb1 qb2
1
2 1
q0
q
,
s4 1
þ e2 E c
½2
0 X 0
qb1 qb2
q 2 1 1 q 2 1 1 s 0 X0 ,
s 0 X0
þ e2 E c
qb1
qb2
þ O e3 ,
408
ðH:3Þ
Sampling and Market Microstructure Noise
where the ‘‘[2]’’ refers to the sum over the two terms where b1 and b2 are permuted.
The first (and leading) term in Equation (H.3),
1
2 1 2 1 q g V
q g V
q0 q1
0
,
,
¼c
c
qb1
qb2
qb1 qb2
2
1
qg
qV
qg 2 1
qV 1
V 1 þ g 2
,
V þ g 2
¼c
qb1
qb1 qb2
qb2
2
1
2
qg
qV
qh
qg
qV 1 qh
¼c
V 1 þ g 2
,
V 1 þ g 2
qb1
qh qb1 qb1
qh qb1
corresponds to the equally spaced, misspecified noise distribution, situation studied in
Section 6.
The second term, linear in e, is zero since
1
1 q0
q 2 1 1 q0
q 2 1 1 ,
s 0 X0
,E
s 0 X0
¼c
E c
qb2
qb1 qb2
qb1
1
q0
q 2 1
,
s 0 E ½X1
¼c
0
qb1 qb2
¼0
with the first equality following from the bilinearity of c, the second from the fact that the
unconditional expectation over the Dis does not depend on the b parameters, so expectation
and differentiation with respect to b2 can be interchanged, and the third equality from the
fact that E [X] ¼ 0.
To calculate the third term in Equation (H.3), the first of two that are quadratic in e, note
that
1
q0
q 4 1 1 1 ,
s 0 X0 X0
a1 E c
qb1 qb2
1
q0
q 4 1 1 1 ,
s 0 E X0 X 0
¼c
qb1 qb2
1
1 q0
q 4 1
,
s 0 diag 1
¼ D20 var½jc
0 0
qb1 qb2
2 1 q g V
q 4 6 1
2
,
s g V diag V 1 V 1 ,
ðH:4Þ
¼ D0 var½jc
qb2
qb1
with the second equality obtained by replacing E½X1
0 X with its value given in Equation
(G.4), and the third by recalling that 0 ¼ g2 V. The elements (i, j) of the two arguments of c
in Equation (H.4) are
q g 2 vi;j
qg 2 i;j
qvi;j qh
¼
v þ g 2
ni;j ¼
qb1
qb1
qh qb1
and
v
k;l
!
N
X
q
4 6
k;m m;m m;l
¼
s g
v v v
qb2
m¼1
!
4 6 N
N
X
q s g
q X
qh
vk;m vm;m vm;l þ s4 g 6
vk;m vm;m vm;l
¼
qb2 m¼1
qh m¼1
qb2
from which c in Equation (H.4) can be evaluated through the sum given in Equation (D.8).
409
The Review of Financial Studies / v 18 n 2 2005
Summing these terms, we obtain
2 1 q g V
q 4 6 1
,
s g V diag V 1 V 1
c
qb2
qb1
4N ðC1V ð1 hÞ þ C2V C3V Þ C1W 1 h2 þ 2C2W C3W ð1 þ 3hÞ
¼
þ oðN Þ,
ð1 hÞ7 ð1 þ hÞ3
where
qg2
qh
, C2V ¼ g2 , C3V ¼
qb1
qb1
q s4 g 6
qh
¼
, C2W ¼ s4 g 6 , C3W ¼
:
qb2
qb2
C1V ¼
C1W
The fourth and last term in Equation (H.3), also quadratic in e,
q 2 1 1 q 2 1 1 s 0 X0 ,
s 0 X0
a2 E c
qb1
qb2
is obtained by first expressing
q 2 1 1 q 2 1 1 s 0 X0 ,
s 0 X0
c
qb1
qb2
in its sum form and then taking expectations term by term. Letting now
q 2 1 1 q 2 1 1 s 0 X0
, vk;l ¼
s 0 X0
ni;j ¼
qb1
qb2
ij
kl
we recall our definition of c(n, v) given in Equation (D.8) whose unconditional expected
value (over the Dis, i.e., over X) we now need to evaluate in order to obtain a2.
We are thus led to consider four-index tensors lijkl and to define
N
N
1n
X
X
~ ðlÞ 2 lh;h;h;h þ
2lh;hþ1;hþ1;hþ1 2lhþ1;hþ1;h;hþ1
c
h¼1
h¼1
þ lh;h;hþ1;hþ1 þ lhþ1;hþ1;h;h þ 4lh;hþ1;h;hþ1
o
2lhþ1;h;h;h 2lh;h;hþ1;h ,
ðH:5Þ
where lijkl is symmetric in the first two and the last two indices, respectively, that is, lijkl ¼ ljikl
and lijkl ¼ lijlk. In terms of our definition of c in Equation (D.8), it should be noted that
~ ðlÞ when one takes lijkl = ni,jvk,l. The expression we seek is, therefore,
cðn, vÞ ¼ c
q 2 1 1 q 2 1 1 ~ ðlÞ,
s 0 X0 ,
s 0 X0
a2 ¼ E c
¼c
ðH:6Þ
qb1
qb2
where lijkl is taken to be the following expected value
lijkl E ni;j vk;l
"
#
q 2 1 1 q 2 1 1 ¼E
s 0 X0
s 0 X0
qb1
ij qb2
kl
"
#
N
X
1 q
q
2
1
1
2
1
¼E
s 0 ir Xrs 0 sj
s 0 kt Xtu 0 ul
qb1
qb2
r;s;t;u¼1
N
X
q 2 1 1 q 2 1 1 ¼
s 0 ir 0 sj
s 0 kt 0 ul E ½Xrs Xtu qb1
qb2
r;s;t;u¼1
N
X
q 2 1 1 q 2 1 1 s 0 ir 0 rj
s 0 kr 0 rl
¼ D20 var½j
qb
qb2
1
r¼1
410
Sampling and Market Microstructure Noise
with the third equality following from the interchangeability of unconditional expectations
and differentiation with respect to b, and the fourth from the fact that E[Xrs Xtu] 6¼ 0 only
when r ¼ s ¼ t ¼ u, and
E ½Xrr Xrr ¼ D20 var½j:
Thus we have
lijkl ¼ D20 var½j
¼ D20 var½j
N
X
q 2 4 1 1 q 2 4 1 1 s g V ir V
s g V kr V rl
rj qb
qb
1
2
r¼1
N
X
q 2 4 i;r r;j q 2 4 k;r r;l s g v v
s g v v :
qb
qb2
1
r¼1
ðH:7Þ
and
q 2 4 i;r r;j q 2 4 k;r r;l s g v v
s g v v ¼
qb1
qb2
2 4 q s g
qðvi;r vr;j Þ qh
vi;r vr;j þ s2 g 4
qh qb1
qb1
2 4 q s g
q vk;r vr;l qh
vk;r vr;l þ s2 g 4
:
qb2
qb2
qh
Summing these terms, we obtain
~ ðlÞ ¼
c
D20 var½j 2N
ð1 hÞ7 ð1 þ hÞ3 ð1 þ h2 Þ3
ðC1l ð1 h4 Þð2C5l C6l ð1 þ h þ h2 þ 2h3 Þ þ C4l ð1 h4 ÞÞ
þ 2C2l C3l ð2C5l C6l ð1 þ 2h þ 4h2 þ 6h3 þ 5h4 þ 4h5 þ 4h6 Þ
þ C4l ð1 þ h þ h2 þ 2h3 h4 h5 h6 2h7 ÞÞÞ
þ oðNÞ,
where
q s2 g 4
qh
, C2l ¼ s2 g 4 , C3l ¼
,
qb1
qb1
2 4 q s g
qh
¼
, C5l ¼ s2 g4 , C6l ¼
:
qb2
qb2
C1l ¼
C4l
Putting it all together, we have
1
1
q
q1
q0 q1
0
,
,
¼c
þ e2 ða1 ½2 þ a2 Þ þ O e3 :
E c
qb1 qb2
qb1 qb2
Finally, the asymptotic variance of the estimator ð^
s2 , ^a2 Þ is given by
1
2 2
avartrue s
^ , ^a ¼ E ½D D0 S1 D ,
ðH:8Þ
where
h i
hi
hi 1
1
1
Etrue €l ¼ Enormal €l ¼ Enormal l_l_0
N
N
N
F ð0Þ þ e2 F ð2Þ þ O e3
D ¼ D0 ¼ 411
The Review of Financial Studies / v 18 n 2 2005
is given by the expression in the correctly specified case (G.15), with F (0) and F (2) given in
Equations (G.16) and (G.17), respectively. Also, in light of Equation (H.1), we have
h i
hi
1
1
S ¼ Etrue l_l_0 ¼ Etrue €l þ cum4 ½U C ¼ D þ cum4 ½U C,
N
N
where
0 1
1
1
q
q1
q
q1
,
,
E c
E c
B
C
2
2
2
2
qs
qs
qs
qa
1 B
C
C¼
B
1
C
1
A
4N @
q
q
,
E c
qa2
qa2
Cð0Þ þ e2 Cð2Þ þ O e3 :
Since, from Equation (H.3), we have
1
1
q
q1
q0 q1
0
E c
,
,
¼c
þ e2 a1 ½2 þ e2 a2 þ O e3 ,
qb1 qb2
qb1 qb2
1
1
q0 q1
c
; 0 , that is,
it follows that C(0) is the matrix with entries
4N
qb1 qb2
0
1
!
1=2 1=2
D0 6a2 þ D0 s2
D0
D0
1
B
C
3
B s2 ð4a2 þ D0 s2 Þ3
C
2a4 sð4a2 þ D0 s2 Þ3=2
ð4a2 þ D0 s2 Þ
B
C
ð0Þ
þ oð1Þ,
C ¼B
2
! C
1=2 2
2
4
2
B
C
D0 s 6a þ D0 s
2a 16a þ 3D0 s
1
@
A
1
3
3=2
2a8
ð4a2 þ D0 s2 Þ
ð4a2 þ D0 s2 Þ
and
Cð2Þ ¼
with
0
B
1
a1 ½2 ¼ var½jB
@
4N
3=2
2D0
ð4a2 þ D0 s2 Þs
9=2
ð4a2 þ D0 s2 Þ
D0
1
ða1 ½2 þ a2 Þ:
4N
1
3=2
sð40a4 þ 2a2 D0 s2 þ D20 s4 Þ
ð4a2 þ D0 s2 Þð4a2 þ D0 s2 Þ D1=2
0
9=2
C
2a4 ð4a2 þ D0 s2 Þ
C
3=2
A
1=2
8D0 s2 ð4a2 þ D0 s2 Þ D0 sð6a2 þ D0 s2 Þ
a4 ð4a2 þ D0 s2 Þ
9=2
þ oð1Þ,
1
a2
4N
0
3=2 1
3=2 D0 40a8 12a4 D0 2 s4 þD0 4 s8 D0 s 44a6 18a4 D0 s2 þ 7a2 D0 2 s6 þ3D0 3 s6
B
C
3
9=2
B 2sð2a2 þD0 s2 Þ ð4a2 þ D0 s2 Þ9=2
C
ð2a2 þ D0 s2 Þ3 ð4a2 þD0 s2 Þ
C
¼ var½jB
B
C
3=2 3
2
2
2
2
4
@
A
2D0 s 50a þ42a D0 s þ 9D0 s
9=2
3
ð2a2 þ D0 s2 Þ ð4a2 þD0 s2 Þ
þoð1Þ,
It follows from Equation (H.8) that
1
2 2
^ , ^a ¼ E ½D DðD þ cum4 ½U CÞ1 D
avartrue s
1 1
¼ D0 D Id þ cum4 ½U D1 C
¼ D0 Id þ cum4 ½U D1 C D1
¼ D0 Id þ cum4 ½U D1 C D1
2 2
¼ avarnormal s
^ , ^a þ D cum4 ½U D1 CD1 ,
412
Sampling and Market Microstructure Noise
where
2 2
^ , ^a ¼ D0 D1 ¼ Að0Þ þ e2 Að2Þ þ O e3
avarnormal s
is the result given in Theorem 3, namely Equation (53).
The correction term due to the misspecification of the error distribution is determined by
cum4[U] times
1 ð0Þ 2 ð2Þ
ð0Þ 2 ð2Þ
1
F þe F þ O e3
D0 D1 CD1 ¼ D0 F ð0Þ þe2 F ð2Þ þ O e3
C þe C þO e3
h
i1
h ð0Þ i1 ð0Þ 2 ð2Þ
¼ D0 Id e2 F ð0Þ F ð2Þ þO e3
C þ e C þO e3
F
h
i1
h ð0Þ i1
Ide2 F ð0Þ F ð2Þ þO e3
F
h
i1
h
i1
¼ D0 F ð0Þ Cð0Þ F ð0Þ
h
i1
h
i1 h
i1
h
i1
h
i1
þe2 D0 F ð0Þ Cð2Þ F ð0Þ
F ð0Þ F ð2Þ F ð0Þ Cð0Þ F ð0Þ
h
i1
h
i1
h
i1 F ð0Þ Cð0Þ F ð0Þ F ð2Þ F ð0Þ
þO e3
Bð0Þ þe2 Bð2Þ þ O e3 :
The asymptotic variance is then given by
2 2 ð0Þ
^ , ^a ¼ A þ cum4 ½U Bð0Þ þ e2 Að2Þ þ cum4 ½U Bð2Þ þ O e3
avartrue s
with the terms A(0), A(2), B(0) and B(2) given in the statement of the theorem.
Appendix I. Proof of Theorem 5
From
c2 1 ebD
E Yi2 ¼ E w2i þ E u2i ¼ s2 D þ
b
it follows that the estimator (5) has the following expected value
N
2 1 X
E Yi2
E s
^ ¼
T i¼1
c2 1 ebD
N
s2 D þ
¼
T
b
2
bD
c
1
e
¼ s2 þ
bD
2
bc2
2
D þ O D2 :
¼ s þc 2
ðI:1Þ
The estimator’s variance is
"
#
N
X
2
1
2
Yi
var s
^ ¼ 2 var
T
i¼1
¼
N
N X
i1
1 X
2 X
var Yi2 þ 2
cov Yi2 , Yj2 :
2
T i¼1
T i¼1 j¼1
413
The Review of Financial Studies / v 18 n 2 2005
Since the Yi s are normal with mean zero,
2
var Yi2 ¼ 2 var½Yi 2 ¼ 2E Yi2
and for i > j
2
2
cov Yi2 , Yj2 ¼ 2 cov Yi , Yj ¼ 2E ui uj
since
cov Yi , Yj ¼ E Yi Yj ¼ E ðwi þ ui Þ wj þ uj ¼ E ui uj :
Now we have
E ui uj ¼ E ðUti Uti1 Þ Utj Utj1
¼ E Uti Utj E Uti Utj1 E Uti1 Utj þ E Uti1 Utj1
2
c2 1 ebD ebDðij1Þ
¼
2b
so that
2
c2 1 ebD ebDðij1Þ
cov Yi2 , Yj2 ¼ 2 2b
4
c4 e2bDðij1Þ 1 ebD
¼
2b2
!2
and consequently
( 2 2 )
2
c2 1 ebD
1 c4 1 ebD Ne2bD 1 þ e2NbD
2
þ
2N
s
D
þ
var s
^ ¼ 2
2
T
b
b2 ð1 þ ebD Þ
ðI:2Þ
with N ¼ T/D. The RMSE expression follows from Equations (I.1) and (I.2). As in Theorem 1,
these are exact small sample expressions, valid for all (T, D).
References
Abramowitz, M., and I. A. Stegun, 1972, Handbook of Mathematical Functions, Dover, New York.
Aı̈t-Sahalia, Y., 1996, ‘‘Nonparametric Pricing of Interest Rate Derivative Securities,’’ Econometrica, 64,
527–560.
Aı̈t-Sahalia, Y., 2002, ‘‘Maximum-Likelihood Estimation of Discretely-Sampled Diffusions: A ClosedForm Approximation Approach,’’ Econometrica, 70, 223–262.
Aı̈t-Sahalia, Y., and P. A. Mykland, 2003, ‘‘The Effects of Random and Discrete Sampling When
Estimating Continuous-Time Diffusions,’’ Econometrica, 71, 483–549.
Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys, 2001, ‘‘The Distribution of Exchange Rate
Realized Volatility,’’ Journal of the American Statistical Association, 96, 42–55.
Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys, 2003, ‘‘Modeling and Forecasting Realized
Volatility,’’ Econometrica, 71, 579–625.
Bandi, F. M., and P. C. B. Phillips, 2003, ‘‘Fully Nonparametric Estimation of Scalar Diffusion Models,’’
Econometrica, 71, 241–283.
Barndorff-Nielsen, O. E., and N. Shephard, 2002, ‘‘Econometric Analysis of Realized Volatility and Its
Use in Estimating Stochastic Volatility Models,’’ Journal of the Royal Statistical Society B, 64, 253–280.
414
Sampling and Market Microstructure Noise
Bessembinder, H., 1994, ‘‘Bid-Ask Spreads in the Interbank Foreign Exchange Markets,’’ Journal of
Financial Economics, 35, 317–348.
Black, F., 1986, ‘‘Noise,’’ Journal of Finance, 41, 529–543.
Choi, J. Y., D. Salandro, and K. Shastri, 1988, ‘‘On the Estimation of Bid-Ask Spreads: Theory and
Evidence,’’ The Journal of Financial and Quantitative Analysis, 23, 219–230.
Conrad, J., G. Kaul, and M. Nimalendran, 1991, ‘‘Components of Short-Horizon Individual Security
Returns,’’ Journal of Financial Economics, 29, 365–384.
Dahlquist, G., and A. Björck, 1974, Numerical Methods, Prentice-Hall Series in Automatic Computation,
New York.
Delattre, S., and J. Jacod, 1997, ‘‘A Central Limit Theorem for Normalized Functions of the Increments
of a Diffusion Process, in the Presence of Round-Off Errors,’’ Bernoulli, 3, 1–28.
Durbin, J., 1959, ‘‘Efficient Estimation of Parameters in Moving-Average Models,’’ Biometrika, 46,
306–316.
Easley, D., and M. O’Hara, 1992, ‘‘Time and the Process of Security Price Adjustment,’’ Journal of
Finance, 47, 577–605.
Engle, R. F., and J. R. Russell, 1998, ‘‘Autoregressive Conditional Duration: A New Model for
Irregularly Spaced Transaction Data,’’ Econometrica, 66, 1127–1162.
French, K., and R. Roll, 1986, ‘‘Stock Return Variances: The Arrival of Information and the Reaction of
Traders,’’ Journal of Financial Economics, 17, 5–26.
Gençay, R., G. Ballocchi, M. Dacorogna, R. Olsen, and O. Pictet, 2002, ‘‘Real-Time Trading
Models and the Statistical Properties of Foreign Exchange Rates,’’ International Economic Review, 43,
463–491.
Glosten, L. R., 1987, ‘‘Components of the Bid-Ask Spread and the Statistical Properties of Transaction
Prices,’’ Journal of Finance, 42, 1293–1307.
Glosten, L. R., and L. E. Harris, 1988, ‘‘Estimating the Components of the Bid/Ask Spread,’’ Journal of
Financial Economics, 21, 123–142.
Gloter, A., and J. Jacod, 2000, ‘‘Diffusions with Measurement Errors: I — Local Asymptotic Normality
and II — Optimal Estimators,’’ working paper, Universite de Paris VI.
Gottlieb, G., and A. Kalay, 1985, ‘‘Implications of the Discreteness of Observed Stock Prices,’’ Journal of
Finance, 40, 135–153.
Haddad, J., 1995, ‘‘On the Closed Form of the Likelihood Function of the First Order Moving Average
Model,’’ Biometrika, 82, 232–234.
Hamilton, J. D., 1995, Time Series Analysis, Princeton University Press, Princeton, NJ.
Hansen, L. P., and J. A. Scheinkman, 1995, ‘‘Back to the Future: Generating Moment Implications for
Continuous-Time Markov Processes,’’ Econometrica, 63, 767–804.
Harris, L., 1990a, ‘‘Estimation of Stock Price Variances and Serial Covariances from Discrete
Observations,’’ Journal of Financial and Quantitative Analysis, 25, 291–306.
Harris, L., 1990b, ‘‘Statistical Properties of the Roll Serial Covariance Bid/Ask Spread Estimator,’’
Journal of Finance, 45, 579–590.
Hasbrouck, J., 1993, ‘‘Assessing the Quality of a Security Market: A New Approach to Transaction-Cost
Measurement,’’ Review of Financial Studies, 6, 191–212.
Heyde, C. C., 1997, Quasi-Likelihood and its Application, Springer-Verlag, New York.
415
The Review of Financial Studies / v 18 n 2 2005
Jacod, J., 1996, ‘‘La Variation Quadratique du Brownien en Présence d’Erreurs d’Arrondi,’’ Asterisque,
236, 155–162.
Kaul, G., and M. Nimalendran, 1990, ‘‘Price Reversals: Bid-Ask Errors or Market Overreaction,’’
Journal of Financial Economics, 28, 67–93.
Lo, A. W., and A. C. MacKinlay, 1990, ‘‘An Econometric Analysis of Nonsynchronous Trading,’’
Journal of Econometrics, 45, 181–211.
Macurdy, T. E., 1982, ‘‘The Use of Time Series Processes to Model the Error Structure of Earnings in a
Longitudinal Data Analysis,’’ Journal of Econometrics, 18, 83–114.
Madhavan, A., M. Richardson, and M. Roomans, 1997, ‘‘Why Do Security Prices Change?,’’ Review of
Financial Studies, 10, 1035–1064.
McCullagh, P., 1987, Tensor Methods in Statistics, Chapman and Hall, London, UK.
McCullagh, P., and J. Nelder, 1989, Generalized Linear Models (2d ed.). Chapman and Hall,
London, UK.
Merton, R. C., 1980, ‘‘On Estimating the Expected Return on the Market: An Exploratory
Investigation,’’ Journal of Financial Economics, 8, 323–361.
Roll, R., 1984, ‘‘A Simple Model of the Implicit Bid-Ask Spread in an Efficient Market,’’ Journal of
Finance, 39, 1127–1139.
Shaman, P., 1969, ‘‘On the Inverse of the Covariance Matrix of a First Order Moving Average,’’
Biometrika, 56, 595–600.
Sias, R. W., and L. T. Starks, 1997, ‘‘Return Autocorrelation and Institutional Investors,’’ Journal of
Financial Economics, 46, 103–131.
White, H., 1982, ‘‘Maximum Likelihood Estimation of Misspecified Models,’’ Econometrica, 50, 1–25.
Zhang, L., P. A. Mykland, and Y. Aı̈t-Sahalia, 2003, ‘‘A Tale of Two Time Scales: Determining
Integrated Volatility with Noisy High-Frequency Data,’’ forthcoming in the Journal of the American
Statistical Association.
416